Warm Up Solve. 3 x + 2 = 8 2. 3. -2k + 7 = -3

Holt Algebra 1

3-1 Graphing and Writing InequalitiesWarm Up

Solve.1. 3x + 2 = 8

3. -2k + 7 = -3

Holt Algebra 1

3-1 Graphing and Writing Inequalities

Identify solutions of inequalities with one variable.

Write and graph inequalities with one variable.

Objectives

Holt Algebra 1

An inequality is a statement that two quantities are not equal. The quantities are compared by using the following signs:

≤A ≤ BA is less than or

equal to B.

<A < BA is lessthan B.

>A > BA is greater

than B.

≥A ≥ B

A is greaterthan or

equal to B.

≠A ≠ BA is not

equal to B.

A solution of an inequality is any value that makes the inequality true.

Holt Algebra 1

3-1 Graphing and Writing InequalitiesExample 1: Identifying Solutions of Inequalities

Describe the solutions of x – 6 ≥ 4 in words.

When the value of x is a number less than 10, the value of x – 6 is less than 4. When the value of x is 10, the value of x – 6 is equal to 4.

When the value of x is a number greater than 10, the value of x – 6 is greater than 4.It appears that the solutions of x – 6 ≥ 4 are all real numbers greater than or equal to 10.

Solution?–9 4≥

–3–9

No–6 4≥

? 3.9 4 ≥ ? 4 4≥

? 4.1 4≥ ? 6 4≥

xx – 6

x – 6 ≥ 4

0 9.9 10 10.1 12–6 3.9 4 4.1 6

NoNo Yes Yes Yes

Holt Algebra 1

An inequality like 3 + x < 9 has too many solutions to list. You can use a graph on a number line to show all the solutions.

The solutions are shaded and an arrow shows that the solutions continue past those shown on the graph. To show that an endpoint is a solution, draw a solid circle at the number. To show an endpoint is not a solution, draw an empty circle.

Holt Algebra 1

3-1 Graphing and Writing InequalitiesExample 2: Graphing Inequalities

Graph each inequality.A. m ≥

0 1– 2 3 3

B. t < 5(–1 + 3) t < 5(–1 + 3) t < 5(2) t < 10

–4 –2 0 2 4 6 8 10 12 –6 –8

Holt Algebra 1

3-1 Graphing and Writing InequalitiesExample 3: Writing an Inequality from a Graph

Write the inequality shown by each graph.

Use any variable. The arrow points to the left, so use either < or ≤. The empty circle at 2 means that 2 is not a solution, so use <.

Use any variable. The arrow points to the right, so use either > or ≥. The solid circle at –0.5 means that –0.5 is a solution, so use ≥.

x ≥ –0.5

Holt Algebra 1

Reading Math “No more than” means “less than or equal to.” “At least” means “greater than or equal to”.

Holt Algebra 1

3-1 Graphing and Writing InequalitiesExample 4: Application

Ray’s dad told him not to turn on the air conditioner unless the temperature is at least 85°F. Define a variable and write an inequality for the temperatures at which Ray can turn on the air conditioner. Graph the solutions.Let t represent the temperatures at which Ray can turn on the air conditioner.

75 80 85 9070

Turn on the AC when temperature is at least 85°Ft ≥ 85

Draw a solid circle at 85. Shade all numbers greater than 85 and draw an arrow pointing to the right.

Holt Algebra 1

A store’s employees earn at least $8.50 per hour. Define a variable and write an inequality for the amount the employees may earn per hour. Graph the solutions.

Check It Out! Example 4

Let w represent an employee’s wages.

An employee earns at least $8.50w ≥ 8.50

4 6 8 10 12−2 0 2 14 16 188.5

w ≥ 8.5

Holt Algebra 1

3-2 Solving Inequalities by Adding or Subtracting

Warm-Up1. Describe the solutions of 7 < x + 4.

2. Graph h ≥ –4.75

–5 –4.75 –4.5Write the inequality shown by each graph.

4.5. A cell phone plan offers free minutes for no more

than 250 minutes per month. Define a variable and write an inequality for the possible number of

free minutes. Graph the solution.

Holt Algebra 1

Solve one-step inequalities by using addition.Solve one-step inequalities by using subtraction.

Objectives

Holt Algebra 1

Example 1A: Using Addition and Subtraction to SolveInequalities

Solve the inequality and graph the solutions.

x + 12 < 20 x + 12 < 20

–12 –12x + 0 < 8

–10 –8 –6 –4 –2 0 2 4 6 8 10

Holt Algebra 1

d – 5 > –7

+5 +5d + 0 > –2

d > –2

d – 5 > –7

Example 1B: Using Addition and Subtraction to SolveInequalities

–10 –8 –6 –4 –2 0 2 4 6 8 10

Holt Algebra 1

Example 1C: Using Addition and Subtraction to SolveInequalities

0.9 ≥ n – 0.3

+0.3 +0.31.2 ≥ n – 0

1.2 ≥ n

0.9 ≥ n – 0.3

Holt Algebra 1

Example 2: Problem-Solving ApplicationSami has a gift card. She has already used $14 of the of the total value, which was $30. Write, solve, and graph an inequality to show how much more she can spend.

Holt Algebra 1

Make a Plan

Example 2 Continued

Write an inequality.Let g represent the remaining amount of money Sami can spend.

g + 14 ≤ 30

Amount remaining plus $30.is at

mostamount

g + 14 ≤ 30

Holt Algebra 1

g + 14 ≤ 30– 14 – 14

g + 0 ≤ 16g ≤ 16

0 2 4 6 8 10 12 14 16 18 10

Example 2 Continued

Holt Algebra 1

Check It Out! Example 2The Recommended Daily Allowance (RDA) of iron for a female in Sarah’s age group (14-18 years) is 15 mg per day. Sarah has consumed 11 mg of iron today. Write and solve an inequality to show how many more milligrams of iron Sarah can consume without exceeding RDA.

Holt Algebra 1

Make a Plan

Write an inequality.Let x represent the amount of iron Sarah needs to consume.

Amount taken plus 15 mgis at

mostamount needed

11 + x 15

Check It Out! Example 2 Continued

Holt Algebra 1

11 + x 15

0 1 2 3 4 5 6 7 8 9 10

x 4. Sarah can consume 4 mg or less of iron without exceeding the RDA.

–11 –11

Holt Algebra 1

3-3 Solving Inequalities by Multiplying or Dividing

Warm-Up

Solve each inequality and graph the solutions.

1. 13 < x + 7x > 6

2. –6 + h ≥ 15h ≥ 21

3. 6.7 + y ≤ –2.1y ≤ –8.8

4. A certain restaurant has room for 120 customers. On one night, there are 72 customers dining. Write and solve an inequality to show how many more people can eat at the restaurant.

Holt Algebra 1

Solve one-step inequalities by using multiplication.Solve one-step inequalities by using division.

Objectives

Holt Algebra 1

Remember, solving inequalities is similar to solving equations. To solve an inequality that contains multiplication or division, undo the operation by dividing or multiplying both sides of the inequality by the same number.

The following rules show the properties of inequality for multiplying or dividing by a positive number. The rules for multiplying or dividing by a negative number appear later in this lesson.

Holt Algebra 1

Example 1A: Multiplying or Dividing by a Positive Number

Solve the inequality and graph the solutions. 7x > –42

7x > –42>

1x > –6 x > –6

–10 –8 –6 –4 –2 0 2 4 6 8 10

Holt Algebra 1

3(2.4) ≤ 3

7.2 ≤ m (or m ≥ 7.2)

0 2 4 6 8 10 12 14 16 18 20

Example 1B: Multiplying or Dividing by a Positive Number

Holt Algebra 1

r < 16

0 2 4 6 8 10 12 14 16 18 20

Example 1C: Multiplying or Dividing by a Positive Number

Holt Algebra 1

If you multiply or divide both sides of an inequality by a negative number, the resulting inequality is not a true statement. You need to reverse the inequality symbol to make the statement true.

Holt Algebra 1

Caution! Do not change the direction of the inequality symbol just because you see a negative sign. For example, you do not change the symbol when solving 4x < –24.

Holt Algebra 1

Example 2A: Multiplying or Dividing by a Negative Number

Solve the inequality and graph the solutions.–12x > 84

x < –7

–10 –8 –6 –4 –2 0 2 4 6–12–14–7

Holt Algebra 1

16 18 20 22 2410 14 26 28 3012

Example 2B: Multiplying or Dividing by a Negative Number

24 x (or x 24)

Holt Algebra 1

Example 3: Application

$4.30 times number of tubes is at most $20.00.

4.30 • p ≤ 20.00

Jill has a $20 gift card to an art supply store where 4 oz tubes of paint are $4.30 each after tax. What are the possible numbers of tubes that Jill can buy?Let p represent the number of tubes of paint that Jill can buy.

Holt Algebra 1

4.30p ≤ 20.00

p ≤ 4.65…

Since p is multiplied by 4.30, divide both sides by 4.30. The symbol does not change.

Since Jill can buy only whole numbers of tubes, she can buy 0, 1, 2, 3, or 4 tubes of paint.

Example 3 Continued

Holt Algebra 1

Warm-UpSolve each inequality and graph the solutions.1. 8x < –24 x < –3 2. –5x ≥ 30 x ≤ –6

3. x > 20 4. x ≥ 6

5. A soccer coach plans to order more shirts for her team. Each shirt costs $9.85. She has $77 left in her uniform budget. What are the possible number of shirts she can buy?0, 1, 2, 3, 4, 5, 6, or 7 shirts

Holt Algebra 1

3-4 Solving Two-Step and Multi-Step Inequalities

Warm-UpSolve each inequality and graph the solutions.

1. 8x < –24 x < –3 2. –5x ≥ 30 x ≤ –6

3. x > 20 4. x ≥ 6

5. A soccer coach plans to order more shirts for her team. Each shirt costs $9.85. She has $77 left in her uniform budget. What are the possible number of shirts she can buy?0, 1, 2, 3, 4, 5, 6, or 7 shirts

Holt Algebra 1

Solve inequalities that contain more than one operation.

Objective

Holt Algebra 1

Inequalities that contain more than one operation require more than one step to solve. Use inverse operations to undo the operations in the inequality one at a time.

Holt Algebra 1

3-4 Solving Two-Step and Multi-Step InequalitiesExample 1A: Solving Multi-Step Inequalities

45 + 2b > 6145 + 2b > 61

–45 –452b > 16

b > 80 2 4 6 8 10 12 14 16 18 20

Holt Algebra 1

8 – 3y ≥ 298 – 3y ≥ 29–8 –8

–3y ≥ 21

y ≤ –7–10 –8 –6 –4 –2 0 2 4 6 8 10

Example 1B: Solving Multi-Step Inequalities

Holt Algebra 1

Check It Out! Example 1cSolve the inequality and graph the solutions.

1 – 2n ≥ 21–1 –1

–2n ≥ 20

n ≤ –10

–10–20 –12 –8 –4–16 0

Holt Algebra 1

To solve more complicated inequalities, you may first need to simplify the expressions on one or both sides by using the order of operations, combining like terms, or using the Distributive Property.

Holt Algebra 1

Example 2A: Simplifying Before Solving Inequalities

Solve the inequality and graph the solutions.2 – (–10) > –4t

12 > –4t

–3 < t (or t > –3)

–3–10 –8 –6 –4 –2 0 2 4 6 8 10

Holt Algebra 1

Example 2B: Simplifying Before Solving InequalitiesSolve the inequality and graph the solutions.

–4(2 – x) ≤ 8

−4(2 – x) ≤ 8−4(2) − 4(−x) ≤

–8 + 4x ≤ 8+8 +8

4x ≤ 16

x ≤ 4

–10 –8 –6 –4 –2 0 2 4 6 8 10

Holt Algebra 1

Check It Out! Example 2bSolve the inequality and graph the solutions.

3 + 2(x + 4) > 33 + 2(x + 4) > 3

3 + 2x + 8 > 32x + 11 > 3

– 11 – 112x > –8

x > –4

–10 –8 –6 –4 –2 0 2 4 6 8 10

Holt Algebra 1

Check It Out! Example 2cSolve the inequality and graph the solutions.

5 < 3x – 2+2 + 2

7 < 3x

Holt Algebra 1

Check It Out! Example 2c ContinuedSolve the inequality and graph the solutions.

7 < 3x

4 6 82 100

Holt Algebra 1

Example 3: ApplicationTo rent a certain vehicle, Rent-A-Ride charges $55.00 per day with unlimited miles. The cost of renting a similar vehicle at We Got Wheels is $38.00 per day plus $0.20 per mile. For what number of miles is the cost at Rent-A-Ride less than the cost at We Got Wheels?

Let m represent the number of miles. The cost for Rent-A-Ride should be less than that of We Got Wheels.

Cost at Rent-A-Ride

must be less than

daily cost at We Got Wheels

plus$0.20

per mile times # of miles.

55 < 38 + 0.20 m

Holt Algebra 1

85 < m

Since 38 is added to 0.20m, subtract 8 from both sides to undo the addition.

Since m is multiplied by 0.20, divide both sides by 0.20 to undo the multiplication.

Rent-A-Ride costs less when the number of miles is more than 85.

Example 3 Continued

55 < 38 + 0.20m

–38 –3855 < 38 + 0.20m

17 < 0.20m

Holt Algebra 1

Check It Out! Example 3

The average of Jim’s two test scores must be at least 90 to make an A in the class. Jim got a 95 on his first test. What grades can Jim get on his second test to make an A in the class?Let x represent the test score needed. The average score is the sum of each score divided by 2.

First test score

plus second test score

divided by

number of scores

is greater than or equal to

total score

(95 + x) 2 ≥ 90

Holt Algebra 1

The score on the second test must be 85 or higher.

Since 95 is added to x, subtract 95 from both sides to undo the addition.

95 + x ≥ 180–95 –95

x ≥ 85

Since 95 + x is divided by 2, multiply both sides by 2 to undo the division.

Holt Algebra 1

Lesson Quiz: Part ISolve each inequality and graph the solutions.1. 13 – 2x ≥ 21 x ≤ –4

2. –11 + 2 < 3p p > –3

3. 23 < –2(3 – t) t > 7

Holt Algebra 1

Lesson Quiz: Part II

5. A video store has two movie rental plans. Plan A includes a $25 membership fee plus $1.25 for each movie rental. Plan B costs $40 for unlimited movie rentals. For what number of movie rentals is plan B less than plan A? more than 12 movies

Holt Algebra 1

3-5 Solving Inequalities with Variables on Both Sides

Warm UpSolve each equation. 1. 2x = 7x + 15 2.

5. Solve and graph 5(2 – b) > 52.

3. 2(3z + 1) = –2(z + 3) 4. 3(p – 1) = 3p + 2

x = –3

b < –3

–5 –3 –2 –1–4 0–6

3y – 21 = 4 – 2y y = 5z = –1

no solution

Holt Algebra 1

Solve inequalities that contain variable terms on both sides.

Objective

Holt Algebra 1

Example 1A: Solving Inequalities with Variables on Both Sides

Solve the inequality and graph the solutions.y ≤ 4y + 18y ≤ 4y + 18–y –y0 ≤ 3y + 18

–18 – 18–18 ≤ 3y

–6 ≤ y (or y –6)

–10 –8 –6 –4 –2 0 2 4 6 8 10

Holt Algebra 1

4m – 3 < 2m + 6–2m – 2m2m – 3 < + 6 + 3 + 32m < 9

Example 1B: Solving Inequalities with Variables on Both Sides

Holt Algebra 1

Example 2: Business Application

The Home Cleaning Company charges $312 to power-wash the siding of a house plus $12 for each window. Power Clean charges $36 per window, and the price includes power-washing the siding. How many windows must a house have to make the total cost from The Home Cleaning Company less expensive than Power Clean?

Let w be the number of windows.

Holt Algebra 1

Example 2 Continued

312 + 12 • w < 36 • w312 + 12w < 36w

– 12w –12w312 < 24w

13 < wThe Home Cleaning Company is less expensive for houses with more than 13 windows.

To collect the variable terms, subtract 12w from both sides.

Since w is multiplied by 24, divide both sides by 24 to undo the multiplication.

HomeCleaningCompany

siding charge

plus$12 per window

# of windows

is lessthan

PowerClean

cost per window

# ofwindows.times

Holt Algebra 1

Check It Out! Example 2A-Plus Advertising charges a fee of $24 plus $0.10 per flyer to print and deliver flyers. Print and More charges $0.25 per flyer. For how many flyers is the cost at A-Plus Advertising less than the cost of Print and More?

Let f represent the number of flyers printed.

24 + 0.10 • f < 0.25 • f

plus$0.10per

is lessthan

# of flyers.

A-PlusAdvertisingfee of $24

Print and More’s cost

per flyer

# of flyerstimes times

Holt Algebra 1

24 + 0.10f < 0.25f–0.10f –0.10f

24 < 0.15f

160 < f

To collect the variable terms, subtract 0.10f from both sides.

Since f is multiplied by 0.15, divide both sides by 0.15 to undo the multiplication.

More than 160 flyers must be delivered to make A-Plus Advertising the lower cost company.

Holt Algebra 1

You may need to simplify one or both sides of an inequality before solving it. Look for like terms to combine and places to use the Distributive Property.

Holt Algebra 1

Check It Out! Example 3aSolve the inequality and graph the solutions.5(2 – r) ≥ 3(r – 2)

5(2 – r) ≥ 3(r – 2)5(2) – 5(r) ≥ 3(r) + 3(–2)

10 – 5r ≥ 3r – 6+6 +616 − 5r ≥ 3r

+ 5r +5r16 ≥ 8r

–6 –2 0 2–4 4

16 ≥ 8r

2 ≥ r

Holt Algebra 1

There are special cases of inequalities called identities and contradictions.

Holt Algebra 1

3-5 Solving Inequalities with Variables on Both SidesExample 4A: Identities and Contradictions

Solve the inequality.2x – 7 ≤ 5 + 2x2x – 7 ≤ 5 + 2x

–2x –2x–7 ≤ 5

The inequality 2x − 7 ≤ 5 + 2x is an identity. All values of x make the inequality true. Therefore, all real numbers are solutions.

Holt Algebra 1

2(3y – 2) – 4 ≥ 3(2y + 7)

2(3y – 2) – 4 ≥ 3(2y + 7)2(3y) – 2(2) – 4 ≥ 3(2y) + 3(7)

6y – 4 – 4 ≥ 6y + 216y – 8 ≥ 6y + 21

Example 4B: Identities and ContradictionsSolve the inequality.

–6y –6y

–8 ≥ 21No values of y make the inequality true. There are no solutions.

Holt Algebra 1

3-6 Solving Compound InequalitiesWarm-Up

Solve each inequality and graph the solutions.1. t < 5t + 24

2. 5x – 9 ≤ 4.1x – 813. 4b + 4(1 – b) > b – 9

Solve each inequality.

4. 2y – 2 ≥ 2(y + 7)

5. 2(–6r – 5) < –3(4r + 2)

Holt Algebra 1

3-6 Solving Compound Inequalities

Solve compound inequalities with one variable. Graph solution sets of compound inequalities with one variable.

Objectives

Holt Algebra 1

The inequalities you have seen so far are simple inequalities. When two simple inequalities are combined into one statement by the words AND or OR, the result is called a compound inequality.

Holt Algebra 1

3-6 Solving Compound InequalitiesExample 1: Chemistry Application

The pH level of a popular shampoo is between 6.0 and 6.5 inclusive. Write a compound inequality to show the pH levels of this shampoo. Graph the solutions.Let p be the pH level of the shampoo.6.0 is less than

or equal topH level is less than

or equal to 6.5

6.0 ≤ p ≤ 6.56.0 ≤ p ≤ 6.5

5.9 6.1 6.2 6.36.0 6.4 6.5

Holt Algebra 1

In this diagram, oval A represents some integer solutions of x < 10 and oval B represents some integer solutions of x > 0. The overlapping region represents numbers that belong in both ovals. Those numbers are solutions of both x < 10 and x > 0.

Holt Algebra 1

You can graph the solutions of a compound inequality involving AND by using the idea of an overlapping region. The overlapping region is called the intersection and shows the numbers that are solutions of both inequalities.

Holt Algebra 1

3-6 Solving Compound InequalitiesExample 2A: Solving Compound Inequalities Involving

ANDSolve the compound inequality and graph the solutions.

–5 < x + 1 < 2 –5 < x + 1 < 2 –1 – 1 – 1–6 < x < 1

–10 –8 –6 –4 –2 0 2 4 6 8 10

Graph –6 < x.

Graph x < 1.Graph the intersection by

finding where the two graphs overlap.

Holt Algebra 1

3-6 Solving Compound InequalitiesExample 2B: Solving Compound Inequalities Involving

ANDSolve the compound inequality and graph the solutions.

8 < 3x – 1 ≤ 11

8 < 3x – 1 ≤ 11+1 +1 +1

9 < 3x ≤ 12

3 < x ≤ 4

Holt Algebra 1

–5 –4 –3 –2 –1 0 1 2 3 4 5

Graph 3 < x.

Graph x ≤ 4.

Graph the intersection by finding where the two graphs overlap.

Example 2B Continued

Holt Algebra 1

Solve the compound inequality and graph the solutions.

Check It Out! Example 2a

–9 < x – 10 < –5

+10 +10 +10–9 < x – 10 < –5

1 < x < 5

–5 –4 –3 –2 –1 0 1 2 3 4 5

Graph 1 < x.

Graph x < 5.

Graph the intersection by finding where the two graphs overlap.

Holt Algebra 1

In this diagram, circle A represents some integer solutions of x < 0, and circle B represents some integer solutions of x > 10. The combined shaded regions represent numbers that are solutions of either x < 0 or x >10.

Holt Algebra 1

You can graph the solutions of a compound inequality involving OR by using the idea of combining regions. The combine regions are called the union and show the numbers that are solutions of either inequality.

Holt Algebra 1

3-6 Solving Compound InequalitiesExample 3A: Solving Compound Inequalities Involving OR

Solve the inequality and graph the solutions.8 + t ≥ 7 OR 8 + t < 28 + t ≥ 7 OR 8 + t < 2–8 –8 –8 −8

t ≥ –1 OR t < –6

–10 –8 –6 –4 –2 0 2 4 6 8 10

Graph t ≥ –1.

Graph t < –6.

Graph the union by combining the regions.

Holt Algebra 1

3-6 Solving Compound InequalitiesExample 3B: Solving Compound Inequalities Involving OR

Solve the inequality and graph the solutions.4x ≤ 20 OR 3x > 21

4x ≤ 20 OR 3x > 21

x ≤ 5 OR x > 7

0 2 4 6 8 10

Graph x ≤ 5.

Graph x > 7.

–10 –8 –6 –4 –2

Holt Algebra 1

Solve the compound inequality and graph the solutions.

Check It Out! Example 3a

2 +r < 12 OR r + 5 > 192 +r < 12 OR r + 5 > 19–2 –2 –5 –5

r < 10 OR r > 14

–4 –2 0 2 4 6 8 10 12 14 16

Graph r < 10.

Graph r > 14.

Solve each simple inequality.

Holt Algebra 1

3-6 Solving Compound InequalitiesExample 4A: Writing a Compound Inequality from a

GraphWrite the compound inequality shown by the graph.

The shaded portion of the graph is not between two values, so the compound inequality involves OR.On the left, the graph shows an arrow pointing left, so use either < or ≤. The solid circle at –8 means –8 is a solution so use ≤. x ≤ –8On the right, the graph shows an arrow pointing right, so use either > or ≥. The empty circle at 0 means that 0 is not a solution, so use >. x > 0The compound inequality is x ≤ –8 OR x > 0.

Holt Algebra 1

3-6 Solving Compound InequalitiesExample 4B: Writing a Compound Inequality from a Graph

The shaded portion of the graph is between the values –2 and 5, so the compound inequality involves AND.The shaded values are on the right of –2, so use > or ≥. The empty circle at –2 means –2 is not a solution, so use >.m > –2The shaded values are to the left of 5, so use < or ≤. The empty circle at 5 means that 5 is not a solution so use <.m < 5The compound inequality is m > –2 AND m < 5 (or -2 < m < 5).

Write the compound inequality shown by the graph.

Holt Algebra 1

Lesson Quiz: Part I

1. The target heart rate during exercise for a 15 year-old is between 154 and 174 beats per minute inclusive. Write a compound inequality to show the heart rates that are within the target range. Graph the solutions.

154 ≤ h ≤ 174

Holt Algebra 1

Lesson Quiz: Part II

Solve each compound inequality and graph the solutions.2. 2 ≤ 2w + 4 ≤ 12

–1 ≤ w ≤ 4

3. 3 + r > −2 OR 3 + r < −7 r > –5 OR r < –10

Holt Algebra 1

Lesson Quiz: Part III

Write the compound inequality shown by each graph.

x < −7 OR x ≥ 05.

−2 ≤ a < 4

Holt Algebra 1

Warm-Up

1. The target heart rate during exercise for a 15 year-old is between 154 and 174 beats per minute inclusive. Write a compound inequality to show the heart rates that are within the target range. Graph the solutions.

Solve each compound inequality and graph the solutions.2. 2 ≤ 2w + 4 ≤ 12

3. 3 + r > −2 OR 3 + r < −7

Holt Algebra 1

Solve inequalities in one variable involving absolute value expressions.

Objectives

Holt Algebra 1

When an inequality contains an absolute-value expression, it can be rewritten as a compound inequality. The inequality |x| < 5 describes all real number whose distance from 0 is less than 5 units. The solutions can be written as -5 < x < 5.

Holt Algebra 1

3-6 Solving Compound InequalitiesExample: Solving Absolute Value Inequalities involving <.

|x| + 3 < 12

– 3 – 3

|x| < 9

|x + 4| ≤ 2

x + 4 ≤ 2 x + 4 ≥ -2 -4 -4 -4 -4

x ≤ -2 and x ≥ -6

Holt Algebra 1

The inequality |x| > 5 describes all real number whose distance from 0 is greater than 5 units. The solutions are all numbers less than -5 or greater than 5. The solutions can be written as x < -5 OR x > 5.

Holt Algebra 1

3-6 Solving Compound InequalitiesExample: Solving Absolute Value Inequalities involving >.

|x| - 20 > -13

+ 20 +20

|x| > 7

|x - 8|+5 ≥ 11

x - 8 ≥ 6 x - 8 ≤ -6 +8 +8 +8 +8

x ≥ 14 or x ≤ 2

- 5 -5|x – 8| ≥ 6

x < -7 or x > 7

Warm Up Solve. 3 x + 2 = 8 2. 3. -2k + 7 = -3

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