6.1 Solve Inequalities Using Addition and Subtraction of the inequality Equivalent inequalities Inequalities that have the same solutions ... LAH_A1_11_FL_NTG_Ch06_148-175.indd 148

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

LAH_A1_11_FL_NTG_Ch06_148-175.indd 148LAH_A1_11_FL_NTG_Ch06_148-175.indd 148 3/2/09 4:24:49 PM3/2/09 4:24:49 PM

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


Remember to use an open circle for < or > and a closed circle for ≤ or ≥.


6.1

Your Notes


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


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


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


Your Notes


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


4. r 1 3 1 } 4 < 5 5. 3 1 m ≥ 7.2

22 121 320

5 76432

Homework


Your Notes


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


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


Your Notes


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 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



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 0, then ac < bc .

If a bc .

If a > b and c > 0, then ac > bc .

If a > b and c < 0, then ac < bc .


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


Your Notes



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.


212 211 210 29 28 27 26 25 24


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



Your Notes


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.


212 211 210 29 28 27 26 25 24


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



Your Notes


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 0, then .

If a b and c > 0, then .

If a > b and c < 0, then .


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.


213 212 211 210 29 28 27 26 25

Example 3 Solve an inequality using division

24x 36


Your Notes


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 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 Use division property of inequality:Divide each side by 24 and reverse the inequality symbol.

x > 29 Simplify.


213 212 211 210 29 28 27 26 25

Example 3 Solve an inequality using division

24x

24

36

24


Your Notes


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.



Your Notes


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.



Your Notes


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 .


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.


228 227 226 225 224 223 222 221 220

Example 2 Solve a multi-step inequality



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.


228 227 226 225 224 223 222 221 220

Example 2 Solve a multi-step inequality



1. 25w 2 2 ≥ 23 2. 2(y 2 2.2) > 0

28 2324252627

3 54210


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


Your Notes


1. 25w 2 2 ≥ 23 2. 2(y 2 2.2) > 0

w ≤ 25 y > 2.2

28 2324252627

3 54210


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


Your Notes


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


Your Notes


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


Your Notes


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


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


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


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


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



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



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



Your Notes


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


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



Your Notes


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


Homework


Your Notes


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


Homework


Your Notes


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


Your Notes


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


Your Notes


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.


Your Notes


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.


Your Notes


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.



Your Notes


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



Your Notes


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


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


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


a. b. y

x

11

y

x

11


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


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|


a. x 22 21 0 1 2g(x) 24

b. x 24 22 0 2 4g(x) 3


a. y

x

11

b. y

x

11


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


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|


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


a. y

x

11

b. y

x

11


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


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 ≥.


Your Notes


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


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 ≥.


Your Notes


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


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



Your Notes


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


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



Your Notes


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



Your Notes


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



Your Notes


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):


3( ) 2 4( ) > 9 Substitute for x and for y.

> 9 Simplify.

(2, 21) a solution.

Example 1 Check solutions of a linear inequality


Your Notes


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):


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):


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


Your Notes


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


Your Notes


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


Your Notes



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



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


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.


Homework

Your Notes

LAH_A1_11_FL_NTG_Ch06_148-175.indd 174LAH_A1_11_FL_NTG_Ch06_148-175.indd 174 3/11/09 12:51:44 AM3/11/09 12:51:44 AM

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.


Homework

Your Notes

LAH_A1_11_FL_NTG_Ch06_148-175.indd 174LAH_A1_11_FL_NTG_Ch06_148-175.indd 174 3/11/09 12:51:44 AM3/11/09 12:51:44 AM

Graph of an inequality

Compound inequality

Absolute deviation

Graph of a 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.


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


t 2 3 > 7 and t > 10


⏐x 1 6⏐ 5 11


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.

Documents

6.1 Solve Inequalities Using Addition and Subtraction of the inequality Equivalent inequalities Inequalities that have the same solutions ... LAH_A1_11_FL_NTG_Ch06_148-175.indd 148