2 the real line, inequalities and comparative phrases

Inequalities

We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left.

Inequalities

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

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

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

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..

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..

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.

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.

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.

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.

Example A. 2 < 4, –3< –2, 0 > –1 are true statementsInequalities

Example A. 2 < 4, –3< –2, 0 > –1 are true statementsand –2 < –5 , 5 < 3 are false statements.

Inequalities

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".

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).

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

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

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

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.

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

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

Example B.a. Draw –1 < x < 3.

Inequalities

Example B.a. Draw –1 < x < 3.

Inequalities

It’s in the natural form.

Example B.a. Draw –1 < x < 3.

Inequalities

It’s in the natural form. Mark the numbers and x on the linein order accordingly.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Example C. Solve x – 3 < 12 and draw the solution.

Inequalities

Example C. Solve x – 3 < 12 and draw the solution.

x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3

Inequalities

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

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

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.

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,

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

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 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,

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

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.

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.

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.

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.

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.

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

A number c is negative means c < 0.

Inequalities

A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign,

Inequalities

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

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

To solve inequalities:1. Simplify both sides of the inequalities

Inequalities

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

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

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

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

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

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

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

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

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

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>

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

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

Example G. Solve 3(2 – x) > 2(x + 9) – 2x

Inequalities

Example G. Solve 3(2 – x) > 2(x + 9) – 2x

3(2 – x) > 2(x + 9) – 2x simplify each side

Inequalities

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

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

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

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

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

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

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

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

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

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

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

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

Example H. (Interval Inequality)

Solve 12 > –2x + 6 > –4 and draw

Inequalities

Example H. (Interval Inequality)

Solve 12 > –2x + 6 > –4 and draw.

12 > –2x + 6 > –4 subtract 6 –6 –6 –6

Inequalities

Example H. (Interval Inequality)

Solve 12 > –2x + 6 > –4 and draw.

12 > –2x + 6 > –4 subtract 6 –6 –6 –6 6 > –2x > –10

Inequalities

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

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

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

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

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

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

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

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

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

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

Inequalities and Comparative PhrasesNon–Positive vs. Non–Negative

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

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

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

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

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

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

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

“ No more/greater than” vs “No less/smaller than”Inequalities and Comparative Phrases

“ 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

“ 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

“ 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

“ 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

“ 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

“ 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

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

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

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

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