2 the real line, inequalities and comparative phrases
TRANSCRIPT
![Page 1: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/1.jpg)
Inequalities
![Page 2: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/2.jpg)
We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left.
Inequalities
![Page 3: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/3.jpg)
We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left.
-2 20 1 3+
-1-3–
Inequalities
![Page 4: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/4.jpg)
We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left.
-2 20 1 3+
-1-3–
2/3
Inequalities
![Page 5: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/5.jpg)
We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left.
-2 20 1 3+
-1-3–
2/3 2½
Inequalities
![Page 6: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/6.jpg)
We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left.
-2 20 1 3+
-1-3–
2/3 2½ π 3.14..
Inequalities
–π –3.14..
![Page 7: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/7.jpg)
We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left.
-2 20 1 3+
-1-3–
2/3 2½ π 3.14..
This line with each position addressed by a real number is called the real (number) line.
Inequalities
–π –3.14..
![Page 8: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/8.jpg)
We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left.
-2 20 1 3+
-1-3–
2/3 2½ π 3.14..
This line with each position addressed by a real number is called the real (number) line.
Inequalities
–π –3.14..
Given two numbers corresponding to two points on the real line, we define the number to the right to be greater than the number to the left.
![Page 9: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/9.jpg)
We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left.
-2 20 1 3+
-1-3–
2/3 2½ π 3.14..
This line with each position addressed by a real number is called the real (number) line.
Inequalities
+–RL
–π –3.14..
Given two numbers corresponding to two points on the real line, we define the number to the right to be greater than the number to the left.
![Page 10: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/10.jpg)
We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left.
-2 20 1 3+
-1-3–
2/3 2½ π 3.14..
This line with each position addressed by a real number is called the real (number) line.
Inequalities
+–R
We write this as L < R and called this the natural form because it corresponds to their respective positions on the real line.
L <
–π –3.14..
Given two numbers corresponding to two points on the real line, we define the number to the right to be greater than the number to the left.
![Page 11: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/11.jpg)
We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left.
-2 20 1 3+
-1-3–
2/3 2½ π 3.14..
This line with each position addressed by a real number is called the real (number) line.
Inequalities
+–R
We write this as L < R and called this the natural form because it corresponds to their respective positions on the real line. This relation may also be written as R > L (less preferable).
L <
–π –3.14..
Given two numbers corresponding to two points on the real line, we define the number to the right to be greater than the number to the left.
![Page 12: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/12.jpg)
Example A. 2 < 4, –3< –2, 0 > –1 are true statementsInequalities
![Page 13: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/13.jpg)
Example A. 2 < 4, –3< –2, 0 > –1 are true statementsand –2 < –5 , 5 < 3 are false statements.
Inequalities
![Page 14: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/14.jpg)
Example A. 2 < 4, –3< –2, 0 > –1 are true statementsand –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them as "all number x where 5 < x".
![Page 15: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/15.jpg)
Example A. 2 < 4, –3< –2, 0 > –1 are true statementsand –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them as "all number x where 5 < x". In general, we write "a < x" for all the numbers x greater than a (excluding a).
![Page 16: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/16.jpg)
Example A. 2 < 4, –3< –2, 0 > –1 are true statementsand –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them as "all number x where 5 < x". In general, we write "a < x" for all the numbers x greater than a (excluding a). In picture,
+–a
open dot
a < x
![Page 17: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/17.jpg)
Example A. 2 < 4, –3< –2, 0 > –1 are true statementsand –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them as "all number x where 5 < x". In general, we write "a < x" for all the numbers x greater than a (excluding a). In picture,
+–a
open dot
If we want all the numbers x greater than or equal to a (including a), we write it as a < x.
a < x
![Page 18: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/18.jpg)
Example A. 2 < 4, –3< –2, 0 > –1 are true statementsand –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them as "all number x where 5 < x". In general, we write "a < x" for all the numbers x greater than a (excluding a). In picture,
+–a
open dot
If we want all the numbers x greater than or equal to a (including a), we write it as a < x. In picture
+–a
solid dot
a < x
a < x
![Page 19: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/19.jpg)
Example A. 2 < 4, –3< –2, 0 > –1 are true statementsand –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them as "all number x where 5 < x". In general, we write "a < x" for all the numbers x greater than a (excluding a). In picture,
+–a
open dot
If we want all the numbers x greater than or equal to a (including a), we write it as a < x. In picture
+–a
solid dot
a < x
a < x
The numbers x fit the description a < x < b where a < b are all the numbers x between a and b.
![Page 20: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/20.jpg)
Example A. 2 < 4, –3< –2, 0 > –1 are true statementsand –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them as "all number x where 5 < x". In general, we write "a < x" for all the numbers x greater than a (excluding a). In picture,
+–a
open dot
If we want all the numbers x greater than or equal to a (including a), we write it as a < x. In picture
+–a
solid dot
a < x
a < x
The numbers x fit the description a < x < b where a < b are all the numbers x between a and b.
+–a a < x < b b
![Page 21: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/21.jpg)
Example A. 2 < 4, –3< –2, 0 > –1 are true statementsand –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them as "all number x where 5 < x". In general, we write "a < x" for all the numbers x greater than a (excluding a). In picture,
+–a
open dot
If we want all the numbers x greater than or equal to a (including a), we write it as a < x. In picture
+–a
solid dot
a < x
a < x
The numbers x fit the description a < x < b where a < b are all the numbers x between a and b. A line segment as such is called an interval.
+–a a < x < b b
![Page 22: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/22.jpg)
Example B.a. Draw –1 < x < 3.
Inequalities
![Page 23: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/23.jpg)
Example B.a. Draw –1 < x < 3.
Inequalities
It’s in the natural form.
![Page 24: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/24.jpg)
Example B.a. Draw –1 < x < 3.
Inequalities
It’s in the natural form. Mark the numbers and x on the linein order accordingly.
![Page 25: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/25.jpg)
Example B.a. Draw –1 < x < 3.
0 3+
-1– x
Inequalities
It’s in the natural form. Mark the numbers and x on the linein order accordingly.
![Page 26: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/26.jpg)
Example B.a. Draw –1 < x < 3.
0 3+
-1– x
b. Draw 0 > x > –3
Inequalities
It’s in the natural form. Mark the numbers and x on the linein order accordingly.
![Page 27: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/27.jpg)
Example B.a. Draw –1 < x < 3.
0 3+
-1– x
b. Draw 0 > x > –3
Inequalities
It’s in the natural form. Mark the numbers and x on the linein order accordingly.
Put it in the natural form –3 < x < 0.
![Page 28: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/28.jpg)
Example B.a. Draw –1 < x < 3.
0 3+
-1– x
b. Draw 0 > x > –3
Inequalities
It’s in the natural form. Mark the numbers and x on the linein order accordingly.
Put it in the natural form –3 < x < 0. Then mark the numbers and x in order accordingly.
![Page 29: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/29.jpg)
Example B.a. Draw –1 < x < 3.
0 3+
-1– x
b. Draw 0 > x > –3
0+
-3–
x
Inequalities
It’s in the natural form. Mark the numbers and x on the linein order accordingly.
Put it in the natural form –3 < x < 0. Then mark the numbers and x in order accordingly.
![Page 30: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/30.jpg)
Example B.a. Draw –1 < x < 3.
0 3+
-1– x
b. Draw 0 > x > –3
0+
-3–
x
Expressions such as 2 < x > 3 or 2 < x < –3 do not have any solution.
Inequalities
It’s in the natural form. Mark the numbers and x on the linein order accordingly.
Put it in the natural form –3 < x < 0. Then mark the numbers and x in order accordingly.
![Page 31: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/31.jpg)
Example B.a. Draw –1 < x < 3.
0 3+
-1– x
b. Draw 0 > x > –3
0+
-3–
x
Expressions such as 2 < x > 3 or 2 < x < –3 do not have any solution.
Inequalities
Adding or subtracting the same quantity to both retains the inequality sign,
It’s in the natural form. Mark the numbers and x on the linein order accordingly.
Put it in the natural form –3 < x < 0. Then mark the numbers and x in order accordingly.
![Page 32: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/32.jpg)
Example B.a. Draw –1 < x < 3.
0 3+
-1– x
b. Draw 0 > x > –3
0+
-3–
x
Expressions such as 2 < x > 3 or 2 < x < –3 do not have any solution.
Inequalities
Adding or subtracting the same quantity to both retains the inequality sign, i.e. if a < b, then a ± c < b ± c.
It’s in the natural form. Mark the numbers and x on the linein order accordingly.
Put it in the natural form –3 < x < 0. Then mark the numbers and x in order accordingly.
![Page 33: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/33.jpg)
Example B.a. Draw –1 < x < 3.
0 3+
-1– x
b. Draw 0 > x > –3
0+
-3–
x
Expressions such as 2 < x > 3 or 2 < x < –3 do not have any solution.
Inequalities
For example 6 < 12, then 6 + 3 < 12 + 3.
Adding or subtracting the same quantity to both retains the inequality sign, i.e. if a < b, then a ± c < b ± c.
It’s in the natural form. Mark the numbers and x on the linein order accordingly.
Put it in the natural form –3 < x < 0. Then mark the numbers and x in order accordingly.
![Page 34: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/34.jpg)
Example B.a. Draw –1 < x < 3.
0 3+
-1– x
b. Draw 0 > x > –3
0+
-3–
x
Expressions such as 2 < x > 3 or 2 < x < –3 do not have any solution.
Inequalities
For example 6 < 12, then 6 + 3 < 12 + 3. We use the this fact to solve inequalities.
Adding or subtracting the same quantity to both retains the inequality sign, i.e. if a < b, then a ± c < b ± c.
It’s in the natural form. Mark the numbers and x on the linein order accordingly.
Put it in the natural form –3 < x < 0. Then mark the numbers and x in order accordingly.
![Page 35: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/35.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
Inequalities
![Page 36: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/36.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3
Inequalities
![Page 37: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/37.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15
Inequalities
![Page 38: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/38.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15
0 15+–
Inequalities
x
![Page 39: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/39.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c.
![Page 40: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/40.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign,
![Page 41: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/41.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign, i.e. if 0 < c and a < b
![Page 42: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/42.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign, i.e. if 0 < c and a < b, then ac < bc.
![Page 43: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/43.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign, i.e. if 0 < c and a < b, then ac < bc. For example 6 < 12 is true,
![Page 44: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/44.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign, i.e. if 0 < c and a < b, then ac < bc. For example 6 < 12 is true, then multiplying by 3 3*6 < 3*12
![Page 45: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/45.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign, i.e. if 0 < c and a < b, then ac < bc. For example 6 < 12 is true, then multiplying by 3 3*6 < 3*12 or 18 < 36 is also true.
![Page 46: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/46.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign, i.e. if 0 < c and a < b, then ac < bc.
Example D. Solve 3x > 12 and draw the solution.
For example 6 < 12 is true, then multiplying by 3 3*6 < 3*12 or 18 < 36 is also true.
![Page 47: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/47.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign, i.e. if 0 < c and a < b, then ac < bc.
Example D. Solve 3x > 12 and draw the solution.
3x > 12
For example 6 < 12 is true, then multiplying by 3 3*6 < 3*12 or 18 < 36 is also true.
![Page 48: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/48.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign, i.e. if 0 < c and a < b, then ac < bc.
Example D. Solve 3x > 12 and draw the solution.
3x > 12 divide by 3 and keep the inequality sign 3x/3 > 12/3
For example 6 < 12 is true, then multiplying by 3 3*6 < 3*12 or 18 < 36 is also true.
![Page 49: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/49.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign, i.e. if 0 < c and a < b, then ac < bc.
Example D. Solve 3x > 12 and draw the solution.
3x > 12 divide by 3 and keep the inequality sign 3x/3 > 12/3 x > 4 or 4 < x
For example 6 < 12 is true, then multiplying by 3 3*6 < 3*12 or 18 < 36 is also true.
![Page 50: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/50.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign, i.e. if 0 < c and a < b, then ac < bc.
Example D. Solve 3x > 12 and draw the solution.
3x > 12 divide by 3 and keep the inequality sign 3x/3 > 12/3 x > 4 or 4 < x
40+–
For example 6 < 12 is true, then multiplying by 3 3*6 < 3*12 or 18 < 36 is also true.
x
![Page 51: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/51.jpg)
A number c is negative means c < 0.
Inequalities
![Page 52: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/52.jpg)
A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign,
Inequalities
![Page 53: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/53.jpg)
A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then
Inequalities
![Page 54: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/54.jpg)
A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb .
Inequalities
![Page 55: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/55.jpg)
A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb .
Inequalities
For example 6 < 12 is true.
![Page 56: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/56.jpg)
A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb .
Inequalities
For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true.
![Page 57: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/57.jpg)
A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb .
Inequalities
For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true. Multiplying by –1 switches the left-right positions of the originals.
![Page 58: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/58.jpg)
A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb .
Inequalities
60+–
12<
For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true. Multiplying by –1 switches the left-right positions of the originals.
![Page 59: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/59.jpg)
A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb .
Inequalities
60+–
12–6 <
For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true. Multiplying by –1 switches the left-right positions of the originals.
![Page 60: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/60.jpg)
A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb .
Inequalities
60+–
12–6–12 <<
For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true. Multiplying by –1 switches the left-right positions of the originals.
![Page 61: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/61.jpg)
A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb .
Example E. Solve –x + 2 < 5 and draw the solution.
Inequalities
60+–
12–6–12 <<
For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true. Multiplying by –1 switches the left-right positions of the originals.
![Page 62: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/62.jpg)
A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb .
Example E. Solve –x + 2 < 5 and draw the solution.
–x + 2 < 5
Inequalities
60+–
12–6–12 <<
For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true. Multiplying by –1 switches the left-right positions of the originals.
![Page 63: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/63.jpg)
A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb .
Example E. Solve –x + 2 < 5 and draw the solution.
–x + 2 < 5 subtract 2 from both sides –x < 3
Inequalities
60+–
12–6–12 <<
For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true. Multiplying by –1 switches the left-right positions of the originals.
![Page 64: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/64.jpg)
A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb .
Example E. Solve –x + 2 < 5 and draw the solution.
–x + 2 < 5 subtract 2 from both sides –x < 3 multiply by –1 to get x, reverse the inequality –(–x) > –3 x > –3
Inequalities
60+–
12–6–12 <<
For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true. Multiplying by –1 switches the left-right positions of the originals.
![Page 65: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/65.jpg)
A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb .
Example E. Solve –x + 2 < 5 and draw the solution.
–x + 2 < 5 subtract 2 from both sides –x < 3 multiply by –1 to get x, reverse the inequality –(–x) > –3 x > –3 or –3 < x
Inequalities
60+–
12–6–12 <<
For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true. Multiplying by –1 switches the left-right positions of the originals.
![Page 66: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/66.jpg)
A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb .
Example E. Solve –x + 2 < 5 and draw the solution.
–x + 2 < 5 subtract 2 from both sides –x < 3 multiply by –1 to get x, reverse the inequality –(–x) > –3 x > –3 or –3 < x
0+
-3–
Inequalities
60+–
12–6–12 <<
For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true. Multiplying by –1 switches the left-right positions of the originals.
![Page 67: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/67.jpg)
To solve inequalities:1. Simplify both sides of the inequalities
Inequalities
![Page 68: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/68.jpg)
To solve inequalities:1. Simplify both sides of the inequalities2. Gather the x-terms to one side and the number-terms to the other sides
Inequalities
![Page 69: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/69.jpg)
To solve inequalities:1. Simplify both sides of the inequalities2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule).
Inequalities
![Page 70: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/70.jpg)
To solve inequalities:1. Simplify both sides of the inequalities2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). 3. Multiply or divide to get x.
Inequalities
![Page 71: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/71.jpg)
To solve inequalities:1. Simplify both sides of the inequalities2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). 3. Multiply or divide to get x. If we multiply or divide by negative numbers to both sides, the inequality sign must be turned around.
Inequalities
![Page 72: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/72.jpg)
To solve inequalities:1. Simplify both sides of the inequalities2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). 3. Multiply or divide to get x. If we multiply or divide by negative numbers to both sides, the inequality sign must be turned around. This rule can be avoided by keeping the x-term positive.
Inequalities
![Page 73: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/73.jpg)
To solve inequalities:1. Simplify both sides of the inequalities2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). 3. Multiply or divide to get x. If we multiply or divide by negative numbers to both sides, the inequality sign must be turned around. This rule can be avoided by keeping the x-term positive.
Inequalities
Example F. Solve 3x + 5 > x + 9
![Page 74: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/74.jpg)
To solve inequalities:1. Simplify both sides of the inequalities2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). 3. Multiply or divide to get x. If we multiply or divide by negative numbers to both sides, the inequality sign must be turned around. This rule can be avoided by keeping the x-term positive.
Inequalities
Example F. Solve 3x + 5 > x + 9
3x + 5 > x + 9 move the x and 5, change side-change sign
![Page 75: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/75.jpg)
To solve inequalities:1. Simplify both sides of the inequalities2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). 3. Multiply or divide to get x. If we multiply or divide by negative numbers to both sides, the inequality sign must be turned around. This rule can be avoided by keeping the x-term positive.
Inequalities
Example F. Solve 3x + 5 > x + 9
3x + 5 > x + 9 move the x and 5, change side-change sign3x – x > 9 – 5
![Page 76: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/76.jpg)
To solve inequalities:1. Simplify both sides of the inequalities2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). 3. Multiply or divide to get x. If we multiply or divide by negative numbers to both sides, the inequality sign must be turned around. This rule can be avoided by keeping the x-term positive.
Inequalities
Example F. Solve 3x + 5 > x + 9
3x + 5 > x + 9 move the x and 5, change side-change sign3x – x > 9 – 5 2x > 4
![Page 77: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/77.jpg)
To solve inequalities:1. Simplify both sides of the inequalities2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). 3. Multiply or divide to get x. If we multiply or divide by negative numbers to both sides, the inequality sign must be turned around. This rule can be avoided by keeping the x-term positive.
Inequalities
Example F. Solve 3x + 5 > x + 9
3x + 5 > x + 9 move the x and 5, change side-change sign3x – x > 9 – 5 2x > 4 div. 2
2x 2
4 2>
![Page 78: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/78.jpg)
To solve inequalities:1. Simplify both sides of the inequalities2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). 3. Multiply or divide to get x. If we multiply or divide by negative numbers to both sides, the inequality sign must be turned around. This rule can be avoided by keeping the x-term positive.
Inequalities
Example F. Solve 3x + 5 > x + 9
3x + 5 > x + 9 move the x and 5, change side-change sign3x – x > 9 – 5 2x > 4 div. 2
2x 2
4 2>
x > 2 or 2 < x
![Page 79: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/79.jpg)
To solve inequalities:1. Simplify both sides of the inequalities2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). 3. Multiply or divide to get x. If we multiply or divide by negative numbers to both sides, the inequality sign must be turned around. This rule can be avoided by keeping the x-term positive.
Inequalities
Example F. Solve 3x + 5 > x + 9
3x + 5 > x + 9 move the x and 5, change side-change sign3x – x > 9 – 5 2x > 4 div. 2
20+–
2x 2
4 2>
x > 2 or 2 < x
![Page 80: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/80.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
Inequalities
![Page 81: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/81.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side
Inequalities
![Page 82: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/82.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x
Inequalities
![Page 83: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/83.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18
Inequalities
![Page 84: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/84.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x
Inequalities
![Page 85: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/85.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x – 12 > 3x
Inequalities
![Page 86: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/86.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x – 12 > 3x–12
33x 3
>
div. by 3 (no need to switch >)
Inequalities
![Page 87: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/87.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x – 12 > 3x–12
33x 3
>
–4 > x or x < –4
div. by 3 (no need to switch >)
Inequalities
![Page 88: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/88.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x – 12 > 3x
0+
–12 3
3x 3
>
-4
div. by 3 (no need to switch >)
Inequalities
–4 > x or x < –4
![Page 89: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/89.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x – 12 > 3x
0+
–12 3
3x 3
>
-4
div. by 3 (no need to switch >)
Inequalities
We also have inequalities in the form of intervals.
–4 > x or x < –4
![Page 90: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/90.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x – 12 > 3x
0+
–12 3
3x 3
>
-4
div. by 3 (no need to switch >)
Inequalities
We also have inequalities in the form of intervals. We solve them by +, –, * , / to all three parts of the inequalities.
–4 > x or x < –4
![Page 91: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/91.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x – 12 > 3x
0+
–12 3
3x 3
>
-4
div. by 3 (no need to switch >)
Inequalities
We also have inequalities in the form of intervals. We solve them by +, –, * , / to all three parts of the inequalities. Again, we + or – remove the number term in the middle first,
–4 > x or x < –4
![Page 92: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/92.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x – 12 > 3x
0+
–12 3
3x 3
>
-4
div. by 3 (no need to switch >)
Inequalities
We also have inequalities in the form of intervals. We solve them by +, –, * , / to all three parts of the inequalities. Again, we + or – remove the number term in the middle first, then divide or multiply to get x.
–4 > x or x < –4
![Page 93: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/93.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x – 12 > 3x
0+
–12 3
3x 3
>
-4
div. by 3 (no need to switch >)
Inequalities
We also have inequalities in the form of intervals. We solve them by +, –, * , / to all three parts of the inequalities. Again, we + or – remove the number term in the middle first, then divide or multiply to get x. The answer is an interval of numbers.
–4 > x or x < –4
![Page 94: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/94.jpg)
Example H. (Interval Inequality)
Solve 12 > –2x + 6 > –4 and draw
Inequalities
![Page 95: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/95.jpg)
Example H. (Interval Inequality)
Solve 12 > –2x + 6 > –4 and draw.
12 > –2x + 6 > –4 subtract 6 –6 –6 –6
Inequalities
![Page 96: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/96.jpg)
Example H. (Interval Inequality)
Solve 12 > –2x + 6 > –4 and draw.
12 > –2x + 6 > –4 subtract 6 –6 –6 –6 6 > –2x > –10
Inequalities
![Page 97: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/97.jpg)
Example H. (Interval Inequality)
Solve 12 > –2x + 6 > –4 and draw.
12 > –2x + 6 > –4 subtract 6 –6 –6 –6 6 > –2x > –10 div. by -2, switch inequality sign 6 -2
-2x -2<
-10 -2
<
Inequalities
![Page 98: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/98.jpg)
Example H. (Interval Inequality)
Solve 12 > –2x + 6 > –4 and draw.
12 > –2x + 6 > –4 subtract 6 –6 –6 –6 6 > –2x > –10
-3 < x < 5
div. by -2, switch inequality sign 6 -2
-2x -2<
-10 -2
<
Inequalities
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Example H. (Interval Inequality)
Solve 12 > –2x + 6 > –4 and draw.
12 > –2x + 6 > –4 subtract 6 –6 –6 –6 6 > –2x > –10
0+
-3 < x < 5
5
div. by -2, switch inequality sign 6 -2
-2x -2<
-10 -2
<
-3
Inequalities
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The following adjectives or comparison phrases may be translated into inequalities:“positive”↔“negative”,“non–positive”↔”non–negative”,“more/greater than”↔ “less/smaller than”,“no more/greater than”↔ “no less/smaller than”, “ at least” ↔”at most”,
Inequalities and Comparative Phrases
![Page 101: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/101.jpg)
The following adjectives or comparison phrases may be translated into inequalities:“positive”↔“negative”,“non–positive”↔”non–negative”,“more/greater than”↔ “less/smaller than”,“no more/greater than”↔ “no less/smaller than”, “ at least” ↔”at most”,
Positive vs. Negative”
Inequalities and Comparative Phrases
![Page 102: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/102.jpg)
The following adjectives or comparison phrases may be translated into inequalities:“positive”↔“negative”,“non–positive”↔”non–negative”,“more/greater than”↔ “less/smaller than”,“no more/greater than”↔ “no less/smaller than”, “ at least” ↔”at most”,
A quantity x is positive means that x is more than 0, i.e. 0 < x,
Positive vs. Negative”
Inequalities and Comparative Phrases
![Page 103: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/103.jpg)
The following adjectives or comparison phrases may be translated into inequalities:“positive”↔“negative”,“non–positive”↔”non–negative”,“more/greater than”↔ “less/smaller than”,“no more/greater than”↔ “no less/smaller than”, “ at least” ↔”at most”,
A quantity x is positive means that x is more than 0, i.e. 0 < x,
0
+ x is positive
Positive vs. Negative”
On the real line:
Inequalities and Comparative Phrases
![Page 104: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/104.jpg)
The following adjectives or comparison phrases may be translated into inequalities:“positive”↔“negative”,“non–positive”↔”non–negative”,“more/greater than”↔ “less/smaller than”,“no more/greater than”↔ “no less/smaller than”, “ at least” ↔”at most”,
A quantity x is positive means that x is more than 0, i.e. 0 < x, and that x is negative means that x is less than 0, i.e. x < 0.
0
+– x is negative x is positive
Positive vs. Negative”
On the real line:
Inequalities and Comparative Phrases
![Page 105: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/105.jpg)
The following adjectives or comparison phrases may be translated into inequalities:“positive”↔“negative”,“non–positive”↔”non–negative”,“more/greater than”↔ “less/smaller than”,“no more/greater than”↔ “no less/smaller than”, “ at least” ↔”at most”,
A quantity x is positive means that x is more than 0, i.e. 0 < x, and that x is negative means that x is less than 0, i.e. x < 0.
0
+– x is negative x is positive
Hence “the temperature T is positive” is translated as “0 < T”.
“the account balance A is negative”, is translated as “A < 0”.
Positive vs. Negative”
On the real line:
Inequalities and Comparative Phrases
![Page 106: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/106.jpg)
The following adjectives or comparison phrases may be translated into inequalities:“positive”↔“negative”,“non–positive”↔”non–negative”,“more/greater than”↔ “less/smaller than”,“no more/greater than”↔ “no less/smaller than”, “ at least” ↔”at most”,
A quantity x is positive means that x is more than 0, i.e. 0 < x, and that x is negative means that x is less than 0, i.e. x < 0.
0
+– x is negative x is positive
Hence “the temperature T is positive” is translated as “0 < T”.
“the account balance A is negative”, is translated as “A < 0”.
Positive vs. Negative”
On the real line:
Inequalities and Comparative Phrases
![Page 107: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/107.jpg)
Inequalities and Comparative PhrasesNon–Positive vs. Non–Negative
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A quantity x is non–positive means x is not positive, or “x ≤ 0”, and that x is non–negative means x is not negative, or “0 ≤ x”,
Non–Positive vs. Non–NegativeInequalities and Comparative Phrases
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A quantity x is non–positive means x is not positive, or “x ≤ 0”, and that x is non–negative means x is not negative, or “0 ≤ x”,
0
+
– x is non–positive
x is non–negative
Non–Positive vs. Non–Negative
On the real line:
Inequalities and Comparative Phrases
![Page 110: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/110.jpg)
A quantity x is non–positive means x is not positive, or “x ≤ 0”, and that x is non–negative means x is not negative, or “0 ≤ x”,
0
+
– x is non–positive
x is non–negative
Hence “the temperature T is non–positive” is “T ≤ 0”.“the account balance A is non–negative” is “0 ≤ A”.
Non–Positive vs. Non–Negative
On the real line:
Inequalities and Comparative Phrases
![Page 111: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/111.jpg)
A quantity x is non–positive means x is not positive, or “x ≤ 0”, and that x is non–negative means x is not negative, or “0 ≤ x”,
0
+
– x is non–positive
x is non–negative
Hence “the temperature T is non–positive” is “T ≤ 0”.“the account balance A is non–negative” is “0 ≤ A”.
Non–Positive vs. Non–Negative
On the real line:
“More/greater than” vs “Less/smaller than”
Inequalities and Comparative Phrases
![Page 112: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/112.jpg)
A quantity x is non–positive means x is not positive, or “x ≤ 0”, and that x is non–negative means x is not negative, or “0 ≤ x”,
0
+
– x is non–positive
x is non–negative
Hence “the temperature T is non–positive” is “T ≤ 0”.“the account balance A is non–negative” is “0 ≤ A”.
Non–Positive vs. Non–Negative
On the real line:
“More/greater than” vs “Less/smaller than”Let C be a number, x is greater than C means “C < x”, and that x is less than C means “x < C”.
Inequalities and Comparative Phrases
![Page 113: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/113.jpg)
A quantity x is non–positive means x is not positive, or “x ≤ 0”, and that x is non–negative means x is not negative, or “0 ≤ x”,
0
+
– x is non–positive
x is non–negative
Hence “the temperature T is non–positive” is “T ≤ 0”.“the account balance A is non–negative” is “0 ≤ A”.
Non–Positive vs. Non–Negative
On the real line:
“More/greater than” vs “Less/smaller than”Let C be a number, x is greater than C means “C < x”, and that x is less than C means “x < C”.
C x is less than C
x is more than COn the real line:
Inequalities and Comparative Phrases
![Page 114: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/114.jpg)
A quantity x is non–positive means x is not positive, or “x ≤ 0”, and that x is non–negative means x is not negative, or “0 ≤ x”,
0
+
– x is non–positive
x is non–negative
Hence “the temperature T is non–positive” is “T ≤ 0”.“the account balance A is non–negative” is “0 ≤ A”.
Non–Positive vs. Non–Negative
On the real line:
“More/greater than” vs “Less/smaller than”Let C be a number, x is greater than C means “C < x”, and that x is less than C means “x < C”.
C x is less than C
x is more than C
Hence “the temperature T is more than –5 ” is “–5 < T”.“the account balance A is less than 1,000” is “A < 1,000”.
On the real line:
Inequalities and Comparative Phrases
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“ No more/greater than” vs “No less/smaller than”Inequalities and Comparative Phrases
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“ No more/greater than” vs “No less/smaller than”A quantity x is no more greater than C means “x ≤ C”, and that x is no–less than C means “C ≤ x”,
Inequalities and Comparative Phrases
![Page 117: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/117.jpg)
“ No more/greater than” vs “No less/smaller than”A quantity x is no more greater than C means “x ≤ C”, and that x is no–less than C means “C ≤ x”,
0
+
– x is no more than C
x is no less than COn the real line:C
Inequalities and Comparative Phrases
![Page 118: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/118.jpg)
“ No more/greater than” vs “No less/smaller than”A quantity x is no more greater than C means “x ≤ C”, and that x is no–less than C means “C ≤ x”,
0
+
– x is no more than C
x is no less than C
“The temperature T is no–more than 250o” is “T ≤ 250”.“The account balance A is no–less than 500” is “500 ≤ A”.
On the real line:C
Inequalities and Comparative Phrases
![Page 119: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/119.jpg)
“ No more/greater than” vs “No less/smaller than”A quantity x is no more greater than C means “x ≤ C”, and that x is no–less than C means “C ≤ x”,
0
+
– x is no more than C
x is no less than C
“The temperature T is no–more than 250o” is “T ≤ 250”.“The account balance A is no–less than 500” is “500 ≤ A”.
On the real line:C
“At least” vs “At most”“At least C” is the same as “no less than C” and“at most C” means “no more than C”.
Inequalities and Comparative Phrases
![Page 120: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/120.jpg)
“ No more/greater than” vs “No less/smaller than”A quantity x is no more greater than C means “x ≤ C”, and that x is no–less than C means “C ≤ x”,
0
+
– x is no more than C
x is no less than C
“The temperature T is no–more than 250o” is “T ≤ 250”.“The account balance A is no–less than 500” is “500 ≤ A”.
On the real line:C
“At least” vs “At most”“At least C” is the same as “no less than C” and“at most C” means “no more than C”.
0
+
– x is at most C
x is at least COn the real line:C
Inequalities and Comparative Phrases
![Page 121: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/121.jpg)
“ No more/greater than” vs “No less/smaller than”A quantity x is no more greater than C means “x ≤ C”, and that x is no–less than C means “C ≤ x”,
0
+
– x is no more than C
x is no less than C
“The temperature T is no–more than 250o” is “T ≤ 250”.“The account balance A is no–less than 500” is “500 ≤ A”.
On the real line:C
“At least” vs “At most”“At least C” is the same as “no less than C” and“at most C” means “no more than C”.
0
+
– x is at most C
x is at least C
“The temperature T is at least 250o” is “250o ≤ T”.
“The account balance A is at most than 500” is “A ≤ 500”.
On the real line:C
Inequalities and Comparative Phrases
![Page 122: 2 the real line, inequalities and comparative phrases](https://reader030.vdocuments.mx/reader030/viewer/2022032700/55d18c9abb61eb906f8b4818/html5/thumbnails/122.jpg)
InequalitiesExercise. A. Draw the following Inequalities. Indicate clearly whether the end points are included or not.1. x < 3 2. –5 ≤ x 3. x < –8 4. x ≤ 12 B. Write in the natural form then draw them.5. x ≥ 3 6. –5 > x 7. x ≥ –8 8. x > 12 C. Draw the following intervals, state so if it is impossible.
9. 6 > x ≥ 3 10. –5 < x ≤ 2 11. 1 > x ≥ –8 12. 4 < x ≤ 2
13. 6 > x ≥ 8
14. –5 > x ≤ 2 15. –7 ≤ x ≤ –3 16. –7 ≤ x ≤ –9D. Solve the following Inequalities and draw the solution.17. x + 5 < 3
18. –5 ≤ 2x + 3 19. 3x + 1 < –8 20. 2x + 3 ≤ 12 – x 21. –3x + 5 ≥ 1 – 4x
22. 2(x + 2) ≥ 5 – (x – 1) 23. 3(x – 1) + 2 ≤ – 2x – 9
24. –2(x – 3) > 2(–x – 1) + 3x 25. –(x + 4) – 2 ≤ 4(x – 1)
26. x + 2(x – 3) < 2(x – 1) – 2
27. –2(x – 3) + 3 ≥ 2(x – 1) + 3x + 13
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Inequalities
F. Solve the following interval inequalities.
28. –4 ≤ 2x 29. 7 > 3
–x 30. < –4–xE. Clear the denominator first then solve and draw the solution.
5x 2 3
1 23 2 + ≥ x31. x 4
–3 3
–4 – 1 > x32.
x 2 83 3
45 – ≤ 33. x 8 12
–5 7 1 + > 34.
x 2 3–3 2
3 4
41 – + x35. x 4 6
5 53
–1 – 2 + < x36.
x 12 27 3
6 1
43 – – ≥ x37.
≤
40. – 2 < x + 2 < 5 41. –1 ≥ 2x – 3 ≥ –11
42. –5 ≤ 1 – 3x < 10 43. 11 > –1 – 3x > –7
38. –6 ≤ 3x < 12 39. 8 > –2x > –4
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Exercise. A. Draw the following Inequalities. Translate each inequality into an English phrase. (There might be more than one ways to do it)1. x < 3 2. –5 ≤ x 3. x < –8 4. x ≤ 12 5. x ≥ 3 6. –5 > x 7. x ≥ –8 8. x > 12 Exercise. B. Translate each English phrase into an inequality. Draw the Inequalities.Let P be the number of people on a bus.9. There were at least 50 people on the bus.10. There were no more than 50 people on the bus.11. There were less than 30 people on the bus.12. There were no less than 28 people on the bus.Let T be temperature outside.13. The temperature is no more than –2o.14. The temperature is at least than 35o.15. The temperature is positive.
Inequalities and Comparative Phrases
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Let M be the amount of money I have.16. I have at most $25.17. I have non–positive amount of money.18. I have less than $45.19. I have at least $250.
Let the basement floor number be given as negative number and that F be floor number that we are on.20. We are below the 7th floor.21. We are above the first floor.22. We are not below the 3rd floor basement.24. We are on at least the 45th floor.25. We are between the 4th floor basement and the 10th floor.26. We are in the basement.
Inequalities and Comparative Phrases