5.3 Solve Multi-Step Inequalities You will solve multi- step inequalities. Essential question: How do you solve multi-step inequalities? You will learn

5.3 Solve Multi-Step Inequalities

• You will solve multi-step inequalities.

• Essential question:• How do you solve

multi-step inequalities?

You will learn how to answer this question by using the distributive property and the properties of inequality.

Warm-Up ExercisesSolve a two-step inequalityEXAMPLE 1

3x – 7 < 8 Write original inequality.

3x < 15 Add 7 to each side.

x < 5 Divide each side by 3.

ANSWER

The solutions are all real numbers less than 5. Check by substituting a number less than 5 in the original inequality.

3x – 7 < 8. Graph your solution.Solve

Warm-Up ExercisesSolve a two-step inequalityEXAMPLE 1

CHECK

Write original inequality.

Solution checks.

Substitute 0 for x.?

3(0) – 7 < 8

3x –7 < 8

–7 < 8

Warm-Up ExercisesSolve a multi-step inequalityEXAMPLE 2

Write original inequality.–0.6(x – 5) 15

–0.6x + 3 15 Distributive property

Subtract 3 from each side.

x –20 Divide each side by 0.6. Reverseinequality symbol.

–

Solve –0.6(x – 5) 15.

–0.6x 12

Is -21 a solution of the inequality? Explain.

Warm-Up ExercisesGUIDED PRACTICE for Examples 1 and 2

Solve the inequality. Graph your solution.

2x – 5 23.1. < –

x 14<–ANSWER



–6y + 5 –16.2. < –

y 3.5> –ANSWER


3. –14

( p –12) > –2

p < 20ANSWER


Warm-Up ExercisesSolve a multi-step inequality EXAMPLE 3

6x – 7 > 2x + 17 Write original inequality.

6x > 2x + 24 Add 7 to each side.

4x > 24 Subtract 2x from each side.

x > 6 Divide each side by 4.

ANSWER

The solutions are all real numbers greater than 6.

6x – 7 > 2x + 17. Graph your solution.Solve

Warm-Up ExercisesIdentify the number of solutions of an inequalityEXAMPLE 4

Solve the inequality, if possible.

a. 14x + 5 < 7(2x – 3)

SOLUTION

14x + 5 < 7(2x – 3)a. Write original inequality.

14x + 5 < 14x – 21 Distributive property

5 < –21 Subtract 14x from each side.

ANSWER

There are no solutions because 5 < –21 is false.


12x – 1 > 6(2x – 1) Write original inequality.

Distributive property

Subtract 12x from each side.

12x – 1 > 12x – 6

–1 > –6

ANSWER

All real numbers are solutions because –1 > –6 is true.

b. 12x – 1 > 6(2x – 1)

Warm-Up Exercises

Solve the inequality, if possible. Graph your solution.

GUIDED PRACTICE for Examples 3 and 4

5x – 12 3x – 44.

x 4ANSWER


5. 5(m + 5) < 5m + 17

GUIDED PRACTICE for Examples 3 and 4

ANSWER no solution


Warm-Up ExercisesIdentify the number of solutions of an inequalityEXAMPLE 4GUIDED PRACTICE for Examples 3 and 4

1 – 8s –4(2s – 1)6. < –


all real numbersANSWER

Warm-Up Exercises

Use the sign shown. A gas station charges $.10 less per gallon of gasoline if a customer also gets a car wash. What are the possible amounts (in gallons) of gasoline that you can buy if you also get a car wash and can spend at most $20?

Car Wash

EXAMPLE 5 Solve a multi-step problem

Warm-Up Exercises

SOLUTION


STEP 1

Write a verbal model. Then write an inequality.

Because you are getting a car wash, you will pay $2.09 – $.10 = $1.99 per gallon of gasoline. Let g be the amount (in gallons) of gasoline that you buy.

1.99 +g 8 20<–

Price ofgasoline

(dollars/gallon)

Amount ofgasoline (gallons)

Price ofcar wash (dollars)

Maximumamount (dollars)

+ <•

Warm-Up ExercisesEXAMPLE 5 Solve a multi-step problem

STEP 2

Solve the inequality.

Write inequality.

Subtract 8 from each side.1.99g ≤ 12

1.99g + 8 ≤ 20

Divide each side by 1.99.g ≤ 6.03015. . .

ANSWER

You can buy up to slightly more than 6 gallons of gasoline.

Warm-Up Exercises

CHECK

You can use a table to check the reasonableness of your answer. The table shows that you will pay $19.94 for exactly 6 gallons of gasoline. Because $19.94 is less than $20, it is reasonable to conclude that you can buy slightly more than 6 gallons of gasoline.


Gasoline

(goal)

Total amount spent

(dollars)

0 8.00

1 9.99

2 11.98

3 13.97

4 15.96

5 17.95

6 19.94

Warm-Up ExercisesGUIDED PRACTICE for Example 3 and 4GUIDED PRACTICE for Example 5

WHAT IF? In Example 5, suppose that a car wash costs $9 and gasoline regularly costs $2.19 per gallon. What are the possible amounts (in gallons) of gasoline that you can buy?

7.

g ≤ 5.02 gal ANSWER

Warm-Up ExercisesEXAMPLE 5 Solve a multi-step problemGUIDED PRACTICE for Example 5

8. You are saving money for a summer camp that costs $1800. You have saved $500 so far, and you have 14 more weeks to save the total amount. What are the possible average amounts of money that you can save per week in order to have a total of at least $1800 saved?

Camp Costs

at least $92.86/wkANSWER

• You will solve multi-step inequalities.

• Essential question:• How do you solve

multi-step inequalities?• You can solve multi-step inequalities by applying the same steps used for solving multi-step equations, with the additional step of reversing an inequality symbol if you multiply or divide by a negative number.

Use the distributive property andthe properties of inequality to produce equivalent inequalities.Reverse the direction of theinequality symbol when dividingor multiplying by a negative number.If you reach an inequality thatis false, there are no solutions; ifyou reach an inequality that isalways true, then all real numbers are solutions.

Warm-Up ExercisesDaily Homework Quiz

1. Solve 3(x + 8) < 9. Graph your solution.

ANSWER all real numbers less than –5


3. 5(a – 2)< 5a – 14

ANSWER no solutions because –10 < –14 is false

Solve the inequality, if possible.

2. 6k + 1 > 3(2k – 3)

ANSWER all real numbers because 1 > –9 is true


4. A box of cat treats contains at least 50 treats. So far you have fed your cats 18 treats. If you want the box of treats to last 8 days, what are the possible average numbers of treats you can feed the cats per day?

ANSWER an average of no more than 4 treats per day.

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5.3 Solve Multi-Step Inequalities You will solve multi- step inequalities. Essential question: How do you solve multi-step inequalities? You will learn