6.1 solve inequalities using addition and subtraction of the inequality equivalent inequalities...
TRANSCRIPT
Solve Inequalities UsingAddition and Subtraction
VOCABULARY
Graph of a linear inequality in one variable
Equivalent inequalities
Goal p Solve inequalities using addition and subtraction.
Food Drive Your school wants to collect at least 5000 pounds of food for a food drive. Write and graph an inequality to describe the amount of food that your school hopes to collect.
Solution
Let p represent the . The value of p must
be 5000 pounds. So, an inequality is .
0 600050004000300020001000 7000 8000
Example 1 Write and graph an inequality
1. You must be 16 years old or older to get your driver's license. Write and graph an inequality to describe the ages of people who may get their driver's license.
13 191817161514 20 21
Checkpoint Complete the following exercise.
Remember to use an open circle for < or > and a closed circle for ≤ or ≥.
148 Lesson 6.1 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
6.1
Your Notes
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Solve Inequalities UsingAddition and Subtraction
VOCABULARY
Graph of a linear inequality in one variable The set of points on a number line that represents all solutions of the inequality
Equivalent inequalities Inequalities that have the same solutions
Goal p Solve inequalities using addition and subtraction.
Food Drive Your school wants to collect at least 5000 pounds of food for a food drive. Write and graph an inequality to describe the amount of food that your school hopes to collect.
Solution
Let p represent the number of pounds of food that the school hopes to collect . The value of p must be greater than or equal to 5000 pounds. So, an inequality is p ≥ 5000 .
0 600050004000300020001000 7000 8000
Example 1 Write and graph an inequality
1. You must be 16 years old or older to get your driver's license. Write and graph an inequality to describe the ages of people who may get their driver's license.
a ≥ 16
13 191817161514 20 21
Checkpoint Complete the following exercise.
Remember to use an open circle for < or > and a closed circle for ≤ or ≥.
148 Lesson 6.1 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
6.1
Your Notes
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ADDITION PROPERTY OF INEQUALITY
Words Adding the same number to each side of an inequality produces an
.
Algebra If a > b, then a 1 c > .
If a < b, then a 1 c < .
If a ≥ b, then a 1 c ≥ .
If a ≤ b, then a 1 c ≤ .
Solve n 2 3.5 < 2.5. Graph your solution.
Solution n 2 3.5 < 2.5 Write original inequality.
n 2 3.5 1 < 2.5 1 Use addition property of inequality: Add to each side.
Simplify.
The solutions are all real numbers . Check by substituting a number for n in the original inequality.
0 654321 7 8
Example 2 Solve an inequality using addition
Checkpoint Solve the inequality. Graph your solution.
2. 6 > y 2 3.3 3. z 2 7 ≥ 4
5 86 1097 11
7 108 12119 13
Copyright © Holt McDougal. All rights reserved. Lesson 6.1 • Algebra 1 Notetaking Guide 149
Your Notes
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ADDITION PROPERTY OF INEQUALITY
Words Adding the same number to each side of an inequality produces an equivalent inequality .
Algebra If a > b, then a 1 c > b 1 c .
If a < b, then a 1 c < b 1 c .
If a ≥ b, then a 1 c ≥ b 1 c .
If a ≤ b, then a 1 c ≤ b 1 c .
Solve n 2 3.5 < 2.5. Graph your solution.
Solution n 2 3.5 < 2.5 Write original inequality.
n 2 3.5 1 3.5 < 2.5 1 3.5 Use addition property of inequality: Add 3.5 to each side.
n < 6 Simplify.
The solutions are all real numbers less than 6 . Check by substituting a number less than 6 for n in the original inequality.
0 654321 7 8
Example 2 Solve an inequality using addition
Checkpoint Solve the inequality. Graph your solution.
2. 6 > y 2 3.3 3. z 2 7 ≥ 4
y < 9.3 z ≥ 11
5 86 1097 11
9.3 7 108 12119 13
Copyright © Holt McDougal. All rights reserved. Lesson 6.1 • Algebra 1 Notetaking Guide 149
Your Notes
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SUBTRACTION PROPERTY OF INEQUALITY
Words Subtracting the same number from each side of an inequality produces an
.
Algebra If a > b, then a 2 c > .
If a < b, then a 2 c < .
If a ≥ b, then a 2 c ≥ .
If a ≤ b, then a 2 c ≤ .
Solve 3 ≤ y 1 8. Graph your solution.
Solution 3 ≤ y 1 8 Write original inequality.
3 2 ≤ y 1 8 2 Use subtraction property of inequality: Subtract from each side.
Simplify.
You can rewrite as .The solutions are all real numbers .
28 27 26 25 24 23 22 21 0
Example 3 Solve an inequality using subtraction
Checkpoint Solve the inequality. Graph your solution.
4. r 1 3 1 } 4 < 5 5. 3 1 m ≥ 7.2
22 121 320
5 76432
Homework
150 Lesson 6.1 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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SUBTRACTION PROPERTY OF INEQUALITY
Words Subtracting the same number from each side of an inequality produces an equivalent inequality .
Algebra If a > b, then a 2 c > b 2 c .
If a < b, then a 2 c < b 2 c .
If a ≥ b, then a 2 c ≥ b 2 c .
If a ≤ b, then a 2 c ≤ b 2 c .
Solve 3 ≤ y 1 8. Graph your solution.
Solution 3 ≤ y 1 8 Write original inequality.
3 2 8 ≤ y 1 8 2 8 Use subtraction property of inequality: Subtract 8 from each side.
25 ≤ y Simplify.
You can rewrite 25 ≤ y as y ≥ 25 .The solutions are all real numbers greater than or equal to 25 .
28 27 26 25 24 23 22 21 0
Example 3 Solve an inequality using subtraction
Checkpoint Solve the inequality. Graph your solution.
4. r 1 3 1 } 4 < 5 5. 3 1 m ≥ 7.2
r < 1 3 } 4 m ≥ 4.2
22 121 320
5 76432
Homework
150 Lesson 6.1 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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6.2 Solve Inequalities UsingMultiplication and DivisionGoal p Solve inequalities using multiplication
and division.
MULTIPLICATION PROPERTY OF INEQUALITY
Words Multiplying each side of an inequality by a number produces an
.
Multiplying each side of an inequality by a number and
produces an equivalent inequality.
Algebra If a < b and c > 0, then .
If a < b and c < 0, then .
If a > b and c > 0, then .
If a > b and c < 0, then .
This property is also true for inequalities involving ≤ and ≥.
Solve y }
9 > 3. Graph your solution.
Solution
y } 9 > 3 Write original inequality.
p y } 9 > p 3 Use multiplication property
of inequality: Multiply each side by .
Simplify.
The solutions are all real numbers .
24 302928272625 31 32
Example 1 Solve an inequality using multiplication
Your Notes
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6.2 Solve Inequalities UsingMultiplication and DivisionGoal p Solve inequalities using multiplication
and division.
MULTIPLICATION PROPERTY OF INEQUALITY
Words Multiplying each side of an inequality by a positive number produces an equivalent inequality .
Multiplying each side of an inequality by a negative number and reversing the direction of the inequality symbol produces an equivalent inequality.
Algebra If a < b and c > 0, then ac < bc .
If a < b and c < 0, then ac > bc .
If a > b and c > 0, then ac > bc .
If a > b and c < 0, then ac < bc .
This property is also true for inequalities involving ≤ and ≥.
Solve y }
9 > 3. Graph your solution.
Solution
y } 9 > 3 Write original inequality.
9 p y } 9 > 9 p 3 Use multiplication property
of inequality: Multiply each side by 9 .
y > 27 Simplify.
The solutions are all real numbers greater than 27 .
24 302928272625 31 32
Example 1 Solve an inequality using multiplication
Your Notes
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Solve m } 22
< 5. Graph your solution.
Solution
m } 22 < 5 Write original inequality.
p m } 22 > p 5 Multiply each side by
and the inequality symbol.
Simplify.
The solutions are all real numbers .
212 211 210 29 28 27 26 25 24
Example 2 Solve an inequality using multiplication
1. r } 7 ≤ 6 2. s } 24 > 0.4
36 4238 464440
22.2 22.0 21.8 21.6 21.4 21.2
3. n } 25 ≥ 22 4. w } 6 < 20.8
7 108 12119
25.0 24.9 24.8 24.7 24.6 24.5
Checkpoint Solve the inequality. Graph your solution.
152 Lesson 6.2 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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Solve m } 22
< 5. Graph your solution.
Solution
m } 22 < 5 Write original inequality.
22 p m } 22 > 22 p 5 Multiply each side by 22
and reverse the inequality symbol.
m > 210 Simplify.
The solutions are all real numbers greater than 210 .
212 211 210 29 28 27 26 25 24
Example 2 Solve an inequality using multiplication
1. r } 7 ≤ 6 2. s } 24 > 0.4
r ≤ 42 s < 21.6
36 4238 464440
22.2 22.0 21.8 21.6 21.4 21.2
3. n } 25 ≥ 22 4. w } 6 < 20.8
n ≤ 10 w < 24.8
7 108 12119
25.0 24.9 24.8 24.7 24.6 24.5
Checkpoint Solve the inequality. Graph your solution.
152 Lesson 6.2 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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DIVISION PROPERTY OF INEQUALITY
Words Dividing each side of an inequality by a number produces an
.
Dividing each side of an inequality by a number and
produces an equivalent inequality.
Algebra If a < b and c > 0, then .
If a < b and c < 0, then .
If a > b and c > 0, then .
If a > b and c < 0, then .
This property is also true for inequalities involving ≤ and ≥.
Solve 24x < 36. Graph your solution.
Solution 24x < 36 Write original inequality.
> Use division property of inequality:Divide each side by and
the inequality symbol.
Simplify.
The solutions are all real numbers .
213 212 211 210 29 28 27 26 25
Example 3 Solve an inequality using division
24x 36
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Your Notes
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DIVISION PROPERTY OF INEQUALITY
Words Dividing each side of an inequality by a positive number produces an equivalent inequality .
Dividing each side of an inequality by a negative number and reversing the direction of the inequality symbol produces an equivalent inequality.
Algebra If a < b and c > 0, then a } c < b } c .
If a < b and c < 0, then a } c > b } c .
If a > b and c > 0, then a } c > b } c .
If a > b and c < 0, then a } c < b } c .
This property is also true for inequalities involving ≤ and ≥.
Solve 24x < 36. Graph your solution.
Solution 24x < 36 Write original inequality.
> Use division property of inequality:Divide each side by 24 and reverse the inequality symbol.
x > 29 Simplify.
The solutions are all real numbers greater than 29 .
213 212 211 210 29 28 27 26 25
Example 3 Solve an inequality using division
24x
24
36
24
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Your Notes
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Homework
Pizza Party You have a budget of $45 to buy pizza for a student council meeting. Pizzas cost $7.50 each. Write and solve an inequality to find the possible numbers of pizzas that you can buy.
Solution
Price per pizza (dollars per pizza)
p Number of pizzas (pizzas) ≤
Budget amount (dollars)
p p ≤
Write inequality.
p ≤ Divide each side by .
You can buy at most pizzas.
Example 4 Solve a real-world problem
5. 29k < 36 6. 10n ≥ 140
26 25 24 23 22 21
12 14 16 18 20 22
7. In Example 4, suppose that you had a budget of $50 and each pizza costs $8. Write and solve an inequality to find the possible numbers of pizzas that you can buy.
Checkpoint Solve the inequality. Graph your solution.
154 Lesson 6.2 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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Homework
Pizza Party You have a budget of $45 to buy pizza for a student council meeting. Pizzas cost $7.50 each. Write and solve an inequality to find the possible numbers of pizzas that you can buy.
Solution
Price per pizza (dollars per pizza)
p Number of pizzas (pizzas) ≤
Budget amount (dollars)
7.50 p p ≤ 45
7.50 p p ≤ 45 Write inequality.
p ≤ 6 Divide each side by 7.50 .
You can buy at most 6 pizzas.
Example 4 Solve a real-world problem
5. 29k < 36 6. 10n ≥ 140
k > 24 n ≥ 14
26 25 24 23 22 21
12 14 16 18 20 22
7. In Example 4, suppose that you had a budget of $50 and each pizza costs $8. Write and solve an inequality to find the possible numbers of pizzas that you can buy.
8 p p ≤ 50; p ≤ 6.25; You can buy at most 6 pizzas.
Checkpoint Solve the inequality. Graph your solution.
154 Lesson 6.2 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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6.3 Solve Multi-Step InequalitiesGoal p Solve multi-step inequalities.
Solve 4x 1 6 ≥ 54. Graph your solution.
Solution 4x 1 6 ≥ 54 Write original inequality.
4x ≥ 48 Subtract from each side.
Divide each side by .
The solutions are all real numbers .
8 14131211109 15 16
Example 1 Solve a two-step inequalityYour Notes
Solve 2 1 } 3 (x 1 21) < 2.
Solution
2 1 } 3 (x 1 21) < 2 Write original inequality.
2 1 } 3 x 2 < 2 Distributive property
2 1 } 3 x < Add to each side.
Multiply each side by . the inequality symbol.
The solutions are all real numbers .
228 227 226 225 224 223 222 221 220
Example 2 Solve a multi-step inequality
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6.3 Solve Multi-Step InequalitiesGoal p Solve multi-step inequalities.
Solve 4x 1 6 ≥ 54. Graph your solution.
Solution 4x 1 6 ≥ 54 Write original inequality.
4x ≥ 48 Subtract 6 from each side.
x ≥ 12 Divide each side by 4 .
The solutions are all real numbers greater than or equal to 12 .
8 14131211109 15 16
Example 1 Solve a two-step inequalityYour Notes
Solve 2 1 } 3 (x 1 21) < 2.
Solution
2 1 } 3 (x 1 21) < 2 Write original inequality.
2 1 } 3 x 2 7 < 2 Distributive property
2 1 } 3 x < 9 Add 7 to each side.
x > 227 Multiply each side by 23 . Reverse the inequality symbol.
The solutions are all real numbers greater than 227 .
228 227 226 225 224 223 222 221 220
Example 2 Solve a multi-step inequality
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1. 25w 2 2 ≥ 23 2. 2(y 2 2.2) > 0
28 2324252627
3 54210
Checkpoint Solve the inequality. Graph your solution.
Solve the inequality, if possible.
a. 8x 1 3 > 2(4x 1 1)
b. 3(8b 2 1) ≤ 24b 2 4
Solutiona. 8x 1 3 > 2(4x 1 1) Write original inequality.
8x 1 3 > Distributive property
Subtract from each side.
are solutions because is .
b. 3(8b 2 1) ≤ 24b 2 4 Write original inequality.
≤ 24b 2 4 Distributive property
Subtract from each side.
There are because is .
Example 3 Identify the number of solutions of an inequality
156 Lesson 6.3 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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1. 25w 2 2 ≥ 23 2. 2(y 2 2.2) > 0
w ≤ 25 y > 2.2
28 2324252627
3 54210
Checkpoint Solve the inequality. Graph your solution.
Solve the inequality, if possible.
a. 8x 1 3 > 2(4x 1 1)
b. 3(8b 2 1) ≤ 24b 2 4
Solutiona. 8x 1 3 > 2(4x 1 1) Write original inequality.
8x 1 3 > 8x 1 2 Distributive property
3 > 2 Subtract 8x from each side.
All real numbers are solutions because 3 > 2 is true .
b. 3(8b 2 1) ≤ 24b 2 4 Write original inequality.
24b 2 3 ≤ 24b 2 4 Distributive property
23 ≤ 24 Subtract 24b from each side.
There are no solutions because 23 ≤ 24 is false .
Example 3 Identify the number of solutions of an inequality
156 Lesson 6.3 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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Homework
3. 18 1 4w ≥ 1 } 2 (8w 1 36) 4. 22(3z 2 1) < 1 2 6z
Checkpoint Solve the inequality, if possible.
Cell Phone Your cell phone plan is $35 a month for 1000 minutes. You are charged $.25 per minute for any additional minutes. What are the possible numbers of additional minutes you can use if you want to spend no more than $50 on your monthly cell phone bill?
SolutionThe amount spent on the monthly plan plus additional minutes must be less than or equal to your monthly budget. Let m be the number of additional minutes that you use.
Price per minute
(dollars/min) p
Number of minutes
(minutes) 1
Monthly fee
(dollars) ≤
Monthly budget (dollars)
p m 1 ≤
≤ Write inequality.
m ≤ Subtract from each side.
m ≤ Divide each side by .
You can use an additional per month to keep within your monthly cell phone budget.
Example 4 Solve a multi-step problem
Copyright © Holt McDougal. All rights reserved. Lesson 6.3 • Algebra 1 Notetaking Guide 157
Your Notes
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Homework
3. 18 1 4w ≥ 1 } 2 (8w 1 36) 4. 22(3z 2 1) < 1 2 6z
All real numbers are No solutionsolutions.
Checkpoint Solve the inequality, if possible.
Cell Phone Your cell phone plan is $35 a month for 1000 minutes. You are charged $.25 per minute for any additional minutes. What are the possible numbers of additional minutes you can use if you want to spend no more than $50 on your monthly cell phone bill?
SolutionThe amount spent on the monthly plan plus additional minutes must be less than or equal to your monthly budget. Let m be the number of additional minutes that you use.
Price per minute
(dollars/min) p
Number of minutes
(minutes) 1
Monthly fee
(dollars) ≤
Monthly budget (dollars)
0.25 p m 1 35 ≤ 50
0.25 p m 1 35 ≤ 50 Write inequality.
0.25 m ≤ 15 Subtract 35 from each side.
m ≤ 60 Divide each side by 0.25 .
You can use an additional 60 minutes or fewer per month to keep within your monthly cell phone budget.
Example 4 Solve a multi-step problem
Copyright © Holt McDougal. All rights reserved. Lesson 6.3 • Algebra 1 Notetaking Guide 157
Your Notes
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Solve Linear Inequalities by Graphing
Your Notes
Goal p Use graphs to solve linear inequalities.
SOLVING LINEAR INEQUALITIES GRAPHICALLY
Step 1 Write the inequality as: ax 1 b , 0 , ax 1 b 0, , or .
Step 2 Write the related equation y 5 1 .
Step 3 Graph the equation y 5 .
• The solutions of ax 1 b 0 are the of the points on the graph of y 5 that lie
the x-axis.
• The solutions of ax 1 b 0 are the of the points on the graph of
y 5 that lie .
• If the inequality symbol is or , the of the graph is also a solution.
Example 1 Solve an inequality graphically
Solve 6x + 4 . 10 graphically.
Solution
Step 1 Write the inequality as
ax 1 b 0
Step 2 Write the function.
y 5
Step 3 Graph y 5 .
From the inequality symbol and the graph’s x-intercept of , x .
CHECK
Choose a number and substitute.
6( ) 1 4 . 10 . 10
The solution checks.158 6.3 Focus on Graphing • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
y
x
11
Focus on GraphingUse after Lesson 6.3
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Solve Linear Inequalities by Graphing
Your Notes
Goal p Use graphs to solve linear inequalities.
SOLVING LINEAR INEQUALITIES GRAPHICALLY
Step 1 Write the inequality as: ax 1 b , 0 , ax 1 b # 0, ax 1 b . 0 , or ax 1 b $ 0 .
Step 2 Write the related equation y 5 ax 1 b .
Step 3 Graph the equation y 5 ax 1 b .
• The solutions of ax 1 b $ 0 are the x-coordinates of the points on the graph of y 5 ax 1 b that lie above the x-axis.
• The solutions of ax 1 b # 0 are the x-coordinates of the points on the graph of y 5 ax 1 b that lie below the x-axis .
• If the inequality symbol is # or $ , the x-intercept of the graph is also a solution.
Example 1 Solve an inequality graphically
Solve 6x + 4 . 10 graphically.
Solution
Step 1 Write the inequality as
ax 1 b . 0 6x 2 6 . 0
Step 2 Write the related function.
y 5 6x 2 6
Step 3 Graph y 5 6x 2 6 .
From the inequality symbol . and the graph’s x-intercept of 1 , x . 1 .
CHECK
Choose a number . 1 and substitute.
6( 2 ) 1 4 . 10 16 . 10
The solution checks.158 6.3 Focus on Graphing • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
y
x
11
(1, 0)
Focus on GraphingUse after Lesson 6.3
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Example 2 Approximate a real-world solution
You can rent a video game for $3.99 for a given number of days. Each additional day costs $1.59. Your budget is $15. How many additional days can you afford?
Solution
Step 1 Write a model. Then write an .
Rate for additional days
•
Additional days
1
Cost of initial rental
#
Amount budgeted
• 1 3.99 #
Write the inequality in the form .
1 3.99 # 15
Step 2 Write the related equation y = .
Step 3 Graph the related equation on a graphing calculator. Draw the line you see on the calculator.
The inequality symbol is and the x-intercept is about . Because you cannot rent a game for part of a day, round x to . To stay within your budget, you can afford to rent up to additional days.
Use this viewing window:Xmin 5 21Xmax 5 9Xsci 5 1Ymin 5 212Ymax 5 3Ysci 5 1
1. 0.5x 1 9 $ 17
2. A used bookstore sells a given number of books
for $8.99 and each additional book for $3.69. How many additional books can you buy if you have $25? Use a graphing calculator.
Checkpoint Solve the inequalities graphically.
2
2
y
x
Copyright © Holt McDougal. All rights reserved. 6.3 Focus on Graphing • Algebra 1 Notetaking Guide 159
Homework
Your Notes
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Example 2 Approximate a real-world solution
You can rent a video game for $3.99 for a given number of days. Each additional day costs $1.59. Your budget is $15. How many additional days can you afford?
Solution
Step 1 Write a verbal model. Then write an inequality .
Rate for additional days
•
Additional days
1
Cost of initial rental
#
Amount budgeted
1.59 • x 1 3.99 # 15
Write the inequality in the form ax + b # 0.
1.59x 1 3.99 # 15 1.59x 2 11.01 # 0
Step 2 Write the related equation y = 1.59x 2 11.01.
Step 3 Graph the related equation on a graphing calculator. Draw the line you see on the calculator.
The inequality symbol is # and the x-intercept is about 6.9 . Because you cannot rent a game for part of a day, round x down to 6 . To stay within your budget, you can afford to rent up to 6 additional days.
Use this viewing window:Xmin 5 21Xmax 5 9Xsci 5 1Ymin 5 212Ymax 5 3Ysci 5 1
1. 0.5x 1 9 $ 17
x $ 16
2. A used bookstore sells a given number of books
for $8.99 and each additional book for $3.69. How many additional books can you buy if you have $25? Use a graphing calculator.
4 additional books
Checkpoint Solve the inequalities graphically.
X 6.9 Y 0.039
2
2
y
x
(16, 0)
Copyright © Holt McDougal. All rights reserved. 6.3 Focus on Graphing • Algebra 1 Notetaking Guide 159
Homework
Your Notes
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6.4 Solve Compound InequalitiesGoal p Solve and graph compound inequalities.
VOCABULARY
Compound inequality
Translate the verbal phrase into an inequality. Then graph the inequality.
a. All real numbers that are greater than or equal to 22 and less than 2.
b. All real numbers that are less than or equal to 3 or greater than 6.
c. All real numbers that are greater than 28 and less than or equal to 23.
Solution
a. 22 x 2
24 210212223 3 4
b. x 3 or x 6
654 7210 3 8
c. 28 x 23
210 272829 242526 2223
Example 1 Write and graph compound inequalities
Your Notes
160 Lesson 6.4 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
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6.4 Solve Compound InequalitiesGoal p Solve and graph compound inequalities.
VOCABULARY
Compound inequality A compound inequality consists of two separate inequalities joined by and or or.
Translate the verbal phrase into an inequality. Then graph the inequality.
a. All real numbers that are greater than or equal to 22 and less than 2.
b. All real numbers that are less than or equal to 3 or greater than 6.
c. All real numbers that are greater than 28 and less than or equal to 23.
Solution
a. 22 ≤ x < 2
24 210212223 3 4
b. x ≤ 3 or x > 6
654 7210 3 8
c. 28 < x ≤ 23
210 272829 242526 2223
Example 1 Write and graph compound inequalities
Your Notes
160 Lesson 6.4 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
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Solve 15 ≤ 3x 2 3 < 24. Graph your solution.
SolutionSeparate the compound inequality into two inequalities. Then solve each inequality separately.
15 ≤ 3x 2 3 and 3x 2 3 < 24 Write two inequalities.
≤ 3x and 3x < Add to each expression.
≤ x and x < Divide each expression by .
The compound inequality can be written as . The solutions are all real numbers
and .
987 10 1143 5 6
Example 2 Solve a compound inequality with and
Solve 15 < 27x 1 1 < 50. Graph your solution.
Solution 15 < 27x 1 1 < 50 Write original
inequality.
< 27x < Subtract from each expression.
> x > Divide each expression by and
.
The solutions are all real numbers and .
29 262728 25 222324 21
Example 3 Solve a compound inequality with and
Copyright © Holt McDougal. All rights reserved. Lesson 6.4 • Algebra 1 Notetaking Guide 161
Your Notes
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Solve 15 ≤ 3x 2 3 < 24. Graph your solution.
SolutionSeparate the compound inequality into two inequalities. Then solve each inequality separately.
15 ≤ 3x 2 3 and 3x 2 3 < 24 Write two inequalities.
18 ≤ 3x and 3x < 27 Add 3 to each expression.
6 ≤ x and x < 9 Divide each expression by 3 .
The compound inequality can be written as 6 ≤ x < 9 . The solutions are all real numbers greater than or equal to 6 and less than 9 .
987 10 1143 5 6
Example 2 Solve a compound inequality with and
Solve 15 < 27x 1 1 < 50. Graph your solution.
Solution 15 < 27x 1 1 < 50 Write original
inequality.
14 < 27x < 49 Subtract 1 from each expression.
22 > x > 27 Divide each expression by 27 and reverse both inequality symbols .
The solutions are all real numbers greater than 27 and less than 22 .
29 262728 25 222324 21
Example 3 Solve a compound inequality with and
Copyright © Holt McDougal. All rights reserved. Lesson 6.4 • Algebra 1 Notetaking Guide 161
Your Notes
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Solve 5x 1 6 ≤ 29 or 2x 2 8 > 12. Graph your solution.
Solution5x 1 6 ≤ 29 or 2x 2 8 > 12 Write original
inequality.
5x ≤ or 2x > Use addition or subtraction property of inequality.
x ≤ or x > Use division property of inequality.
The solutions are all real numbers or .
26 24 22 0 12108642
Example 4 Solve a compound inequality with or
1. 23 ≤ 22x 1 1 < 11
26 25 24 23 22 21 3210
2. 9x 1 1 < 217 or 7x 2 12 > 9
24 23 22 21 543210
Checkpoint Solve the inequality. Graph your solution.
Homework
162 Lesson 6.4 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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Solve 5x 1 6 ≤ 29 or 2x 2 8 > 12. Graph your solution.
Solution5x 1 6 ≤ 29 or 2x 2 8 > 12 Write original
inequality.
5x ≤ 215 or 2x > 20 Use addition or subtraction property of inequality.
x ≤ 23 or x > 10 Use division property of inequality.
The solutions are all real numbers less than or equal to 23 or greater than 10.
26 24 22
23
0 12108642
Example 4 Solve a compound inequality with or
1. 23 ≤ 22x 1 1 < 11
2 ≥ x > 25
26 25 24 23 22 21 3210
2. 9x 1 1 < 217 or 7x 2 12 > 9
x < 22 or x > 3
24 23 22 21 543210
Checkpoint Solve the inequality. Graph your solution.
Homework
162 Lesson 6.4 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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6.5 Solve Absolute Value EquationsGoal p Solve absolute value equations.
VOCABULARY
Absolute value equation
Absolute deviation
SOLVING AN ABSOLUTE VALUE EQUATION
The equation ⏐ax 1 b⏐5 c where c ≥ 0 is equivalent to the statement or .
Solve ⏐x 2 9⏐5 2.
Solution
⏐x 2 9⏐5 2 Write original equation.
x 2 9 5 2 or x 2 9 5 22 Rewrite as two equations.
x 5 or x 5 Add to each side.
The solutions are and . Check your solution.
CHECK
⏐x 2 9⏐5 2 ⏐x 2 9⏐5 2 Write original equation.
⏐ 2 9⏐5 2 ⏐ 2 9⏐5 2 Substitute for x.
⏐ ⏐5 2 ⏐ ⏐5 2 Subtract.
✓ ✓ Simplify. Solution checks.
Example 1 Solve an absolute value equation
Copyright © Holt McDougal. All rights reserved. Lesson 6.5 • Algebra 1 Notetaking Guide 163
Your Notes
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6.5 Solve Absolute Value EquationsGoal p Solve absolute value equations.
VOCABULARY
Absolute value equation An equation that contains an absolute value expression
Absolute deviation The absolute deviation of a number x from a given value is the absolute value of the difference of x and the given value.
SOLVING AN ABSOLUTE VALUE EQUATION
The equation ⏐ax 1 b⏐5 c where c ≥ 0 is equivalent to the statement ax 1 b 5 c or ax 1 b 5 2c.
Solve ⏐x 2 9⏐5 2.
Solution
⏐x 2 9⏐5 2 Write original equation.
x 2 9 5 2 or x 2 9 5 22 Rewrite as two equations.
x 5 11 or x 5 7 Add 9 to each side.
The solutions are 11 and 7 . Check your solution.
CHECK
⏐x 2 9⏐5 2 ⏐x 2 9⏐5 2 Write original equation.
⏐ 11 2 9⏐5 2 ⏐ 7 2 9⏐5 2 Substitute for x.
⏐ 2 ⏐5 2 ⏐ 22 ⏐5 2 Subtract.
2 5 2 ✓ 2 5 2 ✓ Simplify. Solution checks.
Example 1 Solve an absolute value equation
Copyright © Holt McDougal. All rights reserved. Lesson 6.5 • Algebra 1 Notetaking Guide 163
Your Notes
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Solve 4⏐2x 1 8⏐1 6 5 30.
Solution
First, rewrite the equation in the form .
4⏐2x 1 8⏐1 6 5 30 Write original equation.
4⏐2x 1 8⏐5 Subtract from each side.
⏐2x 1 8⏐5 Divide each side by .
Next, solve the absolute value equation.
⏐2x 1 8⏐ 5 Write absolute value equation.
2x 1 8 5 or 2x 1 8 5 Rewrite as two equations.
2x 5 or 2x 5 Subtract from each side.
x 5 or x 5 Divide each side by .
Example 2 Rewrite an absolute value equation
1. ⏐x 1 6⏐5 11 2. 3⏐5x 2 10⏐1 6 5 21
Checkpoint Solve the equation.
Remember to check your solutions in the original equation for accuracy.
164 Lesson 6.4 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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Solve 4⏐2x 1 8⏐1 6 5 30.
Solution
First, rewrite the equation in the form ⏐ax 1 b⏐ 5 c .
4⏐2x 1 8⏐1 6 5 30 Write original equation.
4⏐2x 1 8⏐5 24 Subtract 6 from each side.
⏐2x 1 8⏐5 6 Divide each side by 4 .
Next, solve the absolute value equation.
⏐2x 1 8⏐ 5 6 Write absolute value equation.
2x 1 8 5 6 or 2x 1 8 5 26 Rewrite as two equations.
2x 5 22 or 2x 5 214 Subtract 8 from each side.
x 5 21 or x 5 27 Divide each side by 2 .
Example 2 Rewrite an absolute value equation
1. ⏐x 1 6⏐5 11 2. 3⏐5x 2 10⏐1 6 5 21
5 and 217 3 and 1
Checkpoint Solve the equation.
Remember to check your solutions in the original equation for accuracy.
164 Lesson 6.4 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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Solve ⏐7x 2 3⏐1 8 5 5, if possible.
Solution
⏐7x 2 3⏐1 8 5 5 Write original equation.
⏐7x 2 3⏐5 Subtract from each side.
The absolute value of a number is never . So, there are no solutions.
Example 3 Decide if an equation has no solutions
The absolute deviation of x from 10 is 1.8. Find the values of x that satisfy this requirement.
SolutionAbsolute deviation 5⏐x 2 given value⏐
5⏐x 2 ⏐
Write original equation.
5 x 2 or 5 x 2 Rewrite as two equations.
5 x or 5 x Add to each side.
So, x is or .
Example 4 Use absolute deviation
Homework
3. Find the values of x that satisfy the definition of absolute value for a given value of 213.6 and an absolute deviation of 2.8.
Checkpoint Complete the following exercise.
Copyright © Holt McDougal. All rights reserved. Lesson 6.5 • Algebra 1 Notetaking Guide 165
Your Notes
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Solve ⏐7x 2 3⏐1 8 5 5, if possible.
Solution
⏐7x 2 3⏐1 8 5 5 Write original equation.
⏐7x 2 3⏐5 23 Subtract 8 from each side.
The absolute value of a number is never negative . So, there are no solutions.
Example 3 Decide if an equation has no solutions
The absolute deviation of x from 10 is 1.8. Find the values of x that satisfy this requirement.
SolutionAbsolute deviation 5⏐x 2 given value⏐
1.8 5⏐x 2 10 ⏐
1.8 5 ⏐x 2 10⏐ Write original equation.
1.8 5 x 2 10 or 21.8 5 x 2 10 Rewrite as two equations.
11.8 5 x or 8.2 5 x Add 10 to each side.
So, x is 11.8 or 8.2 .
Example 4 Use absolute deviation
Homework
3. Find the values of x that satisfy the definition of absolute value for a given value of 213.6 and an absolute deviation of 2.8.
210.8 and 216.4
Checkpoint Complete the following exercise.
Copyright © Holt McDougal. All rights reserved. Lesson 6.5 • Algebra 1 Notetaking Guide 165
Your Notes
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Goal p Graph absolute value functions.
Graph Absolute Value Functions
THE PARENT FUNCTION FOR ABSOLUTE VALUE FUNCTIONS
x f(x) 5 |x|
22 |22| 5 221 |21| 5
0
1
2
y
x
11
f(x) |x|
166 6.5 Focus on Functions • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Example 1 Graph g(x) 5 |x 2 h| and g(x) 5 |x| 1 k
Graph the functions. Compare the graphs with the graph of f(x) 5 |x|.a. g(x) 5 |x 2 4| b. g(x) 5 |x| 1 3
Step 1 Make a table of values.
a. b.
x 1 2 3 4 5 x 22 21 0 1 2g(x) 3 g(x) 5
Step 2 Graph the function.
a. b. y
x
11
y
x
11
Step 3 Compare graphs of g and f.
The graph of g(x) 5 |x 2 4| is units of f(x) 5 |x|.
The graph of g(x) 5 |x| 1 3 is units f(x) 5 |x|.
The graphs of g(x) are tranlations of the graph f(x) 5 |x|.Translations can be either horizontal or vertical.
Focus On FunctionsUse after Lesson 6.5
Your Notes
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Goal p Graph absolute value functions.
Graph Absolute Value Functions
THE PARENT FUNCTION FOR ABSOLUTE VALUE FUNCTIONS
x f(x) 5 |x|
22 |22| 5 221 |21| 5 1 0 |0| 5 0
1 |1| 5 1
2 |2| 5 2
y
x
11
f(x) |x|
166 6.5 Focus on Functions • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Example 1 Graph g(x) 5 |x 2 h| and g(x) 5 |x| 1 k
Graph the functions. Compare the graphs with the graph of f(x) 5 |x|.a. g(x) 5 |x 2 4| b. g(x) 5 |x| 1 3
Step 1 Make a table of values.
a. b.
x 1 2 3 4 5 x 22 21 0 1 2g(x) 3 2 1 0 1 g(x) 5 4 3 4 5
Step 2 Graph the function.
a. b. y
x
11
y
x
11
Step 3 Compare graphs of g and f.
The graph of g(x) 5 |x 2 4| is 4 units to the right of f(x) 5 |x|.
The graph of g(x) 5 |x| 1 3 is 3 units above f(x) 5 |x|.
The graphs of g(x) are tranlations of the graph f(x) 5 |x|.Translations can be either horizontal or vertical.
Focus On FunctionsUse after Lesson 6.5
Your Notes
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Example 2 Graph g(x) = a|x|
Graph the functions. Compare the graphs with the graph of f(x) 5 |x|.
a. g(x) 5 22 |x| b. g(x) 5 0.75 |x|
Step 1 Make a table of values.
a. x 22 21 0 1 2g(x) 24
b. x 24 22 0 2 4g(x) 3
Step 2 Graph the function.
a. y
x
11
b. y
x
11
Step 3 Compare graphs of g and f.
a. The graph of g(x) 5 −2|x| opens and is than the graph of f(x) = |x|.
b. The graph of g(x) 5 0.75|x| opens and is than the graph of f(x) 5 |x|.
Graphs can be a vertical stretch or a vertical shrink of the graph of f(x) 5 |x|.
Copyright © Holt McDougal. All rights reserved. 6.5 Focus on Functions • Algebra 1 Notetaking Guide 167
Checkpoint Graph the functions. Compare the graph with the graph of f(x) 5 |x|.
1. g(x) 5 |x 1 2| 2. g(x) 5 |x| 2 0.5 3. g(x) 5 3|x|y
x
11
y
x
11
y
x
11
Homework
Your Notes
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Example 2 Graph g(x) = a|x|
Graph the functions. Compare the graphs with the graph of f(x) 5 |x|.
a. g(x) 5 22 |x| b. g(x) 5 0.75 |x|
Step 1 Make a table of values.
a. x 22 21 0 1 2g(x) 24 22 0 22 24
b. x 24 22 0 2 4g(x) 3 1.5 0 1.5 3
Step 2 Graph the function.
a. y
x
11
b. y
x
11
Step 3 Compare graphs of g and f.
a. The graph of g(x) 5 −2|x| opens down and is narrower than the graph of f(x) = |x|.
b. The graph of g(x) 5 0.75|x| opens up and is wider than the graph of f(x) 5 |x|.
Graphs can be a vertical stretch or a vertical shrink of the graph of f(x) 5 |x|.
Copyright © Holt McDougal. All rights reserved. 6.5 Focus on Functions • Algebra 1 Notetaking Guide 167
Checkpoint Graph the functions. Compare the graph with the graph of f(x) 5 |x|.
1. g(x) 5 |x 1 2| 2. g(x) 5 |x| 2 0.5 3. g(x) 5 3|x|y
x
11
y
x
11
y
x
11
Homework
Your Notes
opens up; 2 units to the left
opens up; narrower
opens up; 0.5 unit down
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6.6 Solve Absolute ValueInequalitiesGoal p Solve absolute value inequalities.
Solve the inequality. Graph your solution.
a. ⏐x⏐ ≤ 9 b. ⏐x⏐ > 1 } 4
Solutiona. The distance between x and 0 is less than or equal
to 9. So, ≤ x ≤ . The solutions are all real numbers and
.
212 630232629 9 12
b. The distance between x and 0 is greater than 1 } 4 .
So, x > or x < . The solutions are all real
numbers or
.
1021
Example 1 Solve an absolute value inequality
SOLVING ABSOLUTE VALUE INEQUALITIES
• The inequality ⏐ax 1 b⏐< c where c > 0 is equivalent to the compound inequality .
• The inequality ⏐ax 1 b⏐> c where c > 0 is equivalent to the compound inequality or
.
Note that < can be replaced by ≤ and > can be replaced by ≥.
168 Lesson 6.6 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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6.6 Solve Absolute ValueInequalitiesGoal p Solve absolute value inequalities.
Solve the inequality. Graph your solution.
a. ⏐x⏐ ≤ 9 b. ⏐x⏐ > 1 } 4
Solutiona. The distance between x and 0 is less than or equal
to 9. So, 29 ≤ x ≤ 9 . The solutions are all real numbers less than or equal to 9 and greater than or equal to 29 .
212 630232629 9 12
b. The distance between x and 0 is greater than 1 } 4 .
So, x > 1 } 4 or x < 2 1 } 4 . The solutions are all real
numbers greater than 1 } 4
or less than 2 1 } 4 .
1021
Example 1 Solve an absolute value inequality
SOLVING ABSOLUTE VALUE INEQUALITIES
• The inequality ⏐ax 1 b⏐< c where c > 0 is equivalent to the compound inequality 2c < ax 1 b < c.
• The inequality ⏐ax 1 b⏐> c where c > 0 is equivalent to the compound inequality ax 1 b < 2c or ax 1 b > c.
Note that < can be replaced by ≤ and > can be replaced by ≥.
168 Lesson 6.6 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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Solve ⏐2x 2 7⏐< 9. Graph your solution.
Solution
⏐2x 2 7⏐< 9 Write original inequality.
< 2x 2 7 < Rewrite as compound inequality.
Add to each expression.
Divide each expression by .
The solutions are all real numbers and . Check several solutions in the original inequality.
864202224 10
Example 2 Solve an absolute value inequality
Solve ⏐x 1 8⏐2 4 ≥ 2. Graph your solution.
Solution
⏐x 1 8⏐2 4 ≥ 2 Write original inequality.
⏐x 1 8⏐≥ Add to each side.
x 1 8 ≥ or x 1 8 ≤ Rewrite as compound inequality.
x ≥ or x ≤ Subtract from each side.
The solutions are all real numbers or .
214 22242628210212
Example 3 Solve an absolute value inequality
Copyright © Holt McDougal. All rights reserved. Lesson 6.6 • Algebra 1 Notetaking Guide 169
Your Notes
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Solve ⏐2x 2 7⏐< 9. Graph your solution.
Solution
⏐2x 2 7⏐< 9 Write original inequality.
29 < 2x 2 7 < 9 Rewrite as compound inequality.
22 < 2x < 16 Add 7 to each expression.
21 < x < 8 Divide each expression by 2 .
The solutions are all real numbers greater than 21 and less than 8 . Check several solutions in the original inequality.
864202224 10
Example 2 Solve an absolute value inequality
Solve ⏐x 1 8⏐2 4 ≥ 2. Graph your solution.
Solution
⏐x 1 8⏐2 4 ≥ 2 Write original inequality.
⏐x 1 8⏐≥ 6 Add 4 to each side.
x 1 8 ≥ 6 or x 1 8 ≤ 26 Rewrite as compound inequality.
x ≥ 22 or x ≤ 214 Subtract 8 from each side.
The solutions are all real numbers greater than or equal to 22 or less than or equal to 214 .
214 22242628210212
Example 3 Solve an absolute value inequality
Copyright © Holt McDougal. All rights reserved. Lesson 6.6 • Algebra 1 Notetaking Guide 169
Your Notes
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Homework
SOLVING INEQUALITIES
One-Step and Multi-Step Inequalities
• Follow the steps for solving an equation, but the inequality symbol when
.
Compound Inequalities
• If necessary, rewrite the inequality as two separate inequalities. Then solve each inequality separately. Include or in the solution.
Absolute Value Inequalities
• If necessary, isolate the absolute value expression on one side of the inequality. Rewrite the absolute value inequality as a . Then solve the compound inequality.
1. 3⏐x 2 6⏐ > 9 2. ⏐6x 2 11⏐ ≤ 7
23 0 3 6 9 12
10 2 3 4 5
3. 22⏐6x 2 1⏐1 5 < 3 121 0
Checkpoint Solve the inequality. Graph your solution.
170 Lesson 6.6 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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Homework
SOLVING INEQUALITIES
One-Step and Multi-Step Inequalities
• Follow the steps for solving an equation, but reverse the inequality symbol when multiplying or dividing by a negative number .
Compound Inequalities
• If necessary, rewrite the inequality as two separate inequalities. Then solve each inequality separately. Include and or or in the solution.
Absolute Value Inequalities
• If necessary, isolate the absolute value expression on one side of the inequality. Rewrite the absolute value inequality as a compound inequality . Then solve the compound inequality.
1. 3⏐x 2 6⏐ > 9 2. ⏐6x 2 11⏐ ≤ 7
x > 9 or x < 3 2 } 3 ≤ x ≤ 3
23 0 3 6 9 12
10 2 3 4 5
23
3. 22⏐6x 2 1⏐1 5 < 3 121 0
x > 1 } 3 or x < 0
Checkpoint Solve the inequality. Graph your solution.
170 Lesson 6.6 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
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6.7
VOCABULARY
Linear inequality in two variables
Graph of an inequality in two variables
Goal p Graph linear inequalities in two variables.
Graph Linear Inequalitiesin Two Variables
Tell whether the ordered pair is a solution of 3x 2 4y > 9.
a. (2, 0) b. (2, 21)
Solutiona. Test (2, 0):
3x 2 4y > 9 Write inequality.
3( ) 2 4( ) > 9 Substitute for x and for y.
> 9 Simplify.
(2, 0) a solution.
b. Test (2, 21):
3x 2 4y > 9 Write inequality.
3( ) 2 4( ) > 9 Substitute for x and for y.
> 9 Simplify.
(2, 21) a solution.
Example 1 Check solutions of a linear inequality
Copyright © Holt McDougal. All rights reserved. Lesson 6.7 • Algebra 1 Notetaking Guide 171
Your Notes
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6.7
VOCABULARY
Linear inequality in two variables A linear inequality in two variables is the result of replacing the 5 sign in a linear equation with <, ≤, >, or ≥.
Graph of an inequality in two variables The set of points that represent all solutions of the inequality
Goal p Graph linear inequalities in two variables.
Graph Linear Inequalitiesin Two Variables
Tell whether the ordered pair is a solution of 3x 2 4y > 9.
a. (2, 0) b. (2, 21)
Solutiona. Test (2, 0):
3x 2 4y > 9 Write inequality.
3( 2 ) 2 4( 0 ) > 9 Substitute 2 for x and 0 for y.
6 > 9 Simplify.
(2, 0) is not a solution.
b. Test (2, 21):
3x 2 4y > 9 Write inequality.
3( 2 ) 2 4( 21 ) > 9 Substitute 2 for x and 21 for y.
10 > 9 Simplify.
(2, 21) is a solution.
Example 1 Check solutions of a linear inequality
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Your Notes
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GRAPHING A LINEAR INEQUALITY IN TWO VARIABLES
Step 1 Graph the boundary line. Use a line for < or >, and use a line for ≤ or ≥.
Step 2 Test a point not on by checking whether the ordered pair is a solution of the inequality.
Step 3 Shade the containing the point if the ordered pair a solution of the inequality. Shade the if the ordered pair a solution.
Graph the inequality y < 2 1 } 2 x 1 4.
Solution
1. Graph the equation y 5 2 1 } 2 x 1 4. The inequality is
<, so use a line.
2. Test (0, 0) in y < 2 1 } 2 x 1 4.
< 2 1 } 2 ( ) 1 4
<
3. the half-plane that (0, 0) because (0, 0) a solution of the inequality.
x
y
2
6
10
22222
26
6 10
Example 2 Graph a linear inequality in two variables
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Your Notes
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GRAPHING A LINEAR INEQUALITY IN TWO VARIABLES
Step 1 Graph the boundary line. Use a dashed line for < or >, and use a solid line for ≤ or ≥.
Step 2 Test a point not on the boundary line by checking whether the ordered pair is a solution of the inequality.
Step 3 Shade the half-plane containing the point if the ordered pair is a solution of the inequality. Shade the other half-plane if the ordered pair is not a solution.
Graph the inequality y < 2 1 } 2 x 1 4.
Solution
1. Graph the equation y 5 2 1 } 2 x 1 4. The inequality is
<, so use a dashed line.
2. Test (0, 0) in y < 2 1 } 2 x 1 4.
0 < 2 1 } 2 ( 0 ) 1 4
0 < 4
3. Shade the half-plane that contains (0, 0) because (0, 0) is a solution of the inequality.
x
y
2
6
10
22222
26
6 10
Example 2 Graph a linear inequality in two variables
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Your Notes
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Copyright © Holt McDougal. All rights reserved. Lesson 6.7 • Algebra 1 Notetaking Guide 173
Graph the inequality x ≥ 4.
Solution 1. Graph the equation x 5 4. The inequality is ≥,
so use a line.
2. Test (0, 3) in x ≥ 4. You only substitute the because the inequality does
not have the variable . ≥ 4
3. the half-plane that (0, 3), because (0, 3) a solution of the inequality.
x
y
1
3
12121
23
3 5
Example 3 Graph a linear inequality in one variable
1. 2y 1 4x > 8 2. y < 2
x
y
1
3
5
1212321
23
3 5
x
y
1
3
1212321
23
3
Checkpoint Graph the inequality.
Your Notes
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Copyright © Holt McDougal. All rights reserved. Lesson 6.7 • Algebra 1 Notetaking Guide 173
Graph the inequality x ≥ 4.
Solution 1. Graph the equation x 5 4. The inequality is ≥,
so use a solid line.
2. Test (0, 3) in x ≥ 4. You only substitute the x-coordinate because the inequality does not have the variable y .
0 ≥ 4
3. Shade the half-plane that does not contain (0, 3), because (0, 3) is not a solution of the inequality.
x
y
1
3
12121
23
3 5
Example 3 Graph a linear inequality in one variable
1. 2y 1 4x > 8 2. y < 2
x
y
1
3
5
1212321
23
3 5
x
y
1
3
1212321
23
3
Checkpoint Graph the inequality.
Your Notes
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Example 4 Write an inequality represented by a graph
Write an inequality represented by the graph.y
x
11
Solution
Step 1 Note that the of the boundary line is
and the -intercept is .
Step 2 Write an equation in slope-intercept form
y 5 . Substitute for and
for .
Step 3 Write an inequality. The boundary line is , so replace 5 with either or . The graph is shaded the boundary line, so choose . The inequality represented by the graph is
3. 4.
y
x11
y
x
11
Checkpoint Write an inequality represented by the graph.
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Homework
Your Notes
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Example 4 Write an inequality represented by a graph
Write an inequality represented by the graph.y
x
11
Solution
Step 1 Note that the slope of the boundary line is 2 1 } 3
and the y -intercept is 2 .
Step 2 Write an equation in slope-intercept form
y 5 mx 1 b . Substitute 2 1 } 3 for m and 2
for b .
y 5 2 1 } 3 x 1 2
Step 3 Write an inequality. The boundary line is solid , so replace 5 with either # or $ . The graph is shaded above the boundary line, so choose $ . The inequality represented by the graph is
y $ 2 1 } 3 x 1 2
3. 4.
y
x11
y
x
11
y , 2 3 } 2 x 23 y $ x 1 1
Checkpoint Write an inequality represented by the graph.
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Homework
Your Notes
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Graph of an inequality
Compound inequality
Absolute deviation
Graph of a linear inequality in two variables
Equivalent inequalities
Absolute value equation
Linear inequality in two variables
Review your notes and Chapter 6 by using the Chapter Review on pages 426–429 of your textbook.
Copyright © Holt McDougal. All rights reserved. Words to Review • Algebra 1 Notetaking Guide 175
Words to ReviewGive an example of the vocabulary word.
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Graph of an inequality
2324 22 121 3 420
Compound inequality
22 ≤ x < 2
Absolute deviation
⏐x 2 given value⏐
Graph of a linear inequality in two variables
x
y
1
3
1212321
23
3 5
Equivalent inequalities
t 2 3 > 7 and t > 10
Absolute value equation
⏐x 1 6⏐ 5 11
Linear inequality in two variables
y < 2x 1 4
Review your notes and Chapter 6 by using the Chapter Review on pages 426–429 of your textbook.
Copyright © Holt McDougal. All rights reserved. Words to Review • Algebra 1 Notetaking Guide 175
Words to ReviewGive an example of the vocabulary word.
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