Review 2.1-2.5Advanced Algebra 1

Word Problems with Inequalities

Don’t panic! Remember to: •Read the questions carefully.•Define the variable.•Write an inequality.•Solve the inequality.•Check that your answer is reasonable.•Answer in a complete sentence.

Example 1Eight less than the product of -3 and a number is greater than -26. Write and solve an inequality to represent this relationship. Graph the solution set, and check your answer.

Let x = the number-3x – 8 > -26

+8 +8 -3x > -18 -3 -3 x < 6

4 5 6 7 8

CHECK @ x = 5-2x – 6 > -18 -2(5) – 6 > -18-10 – 6 > -18-16 > -18 Don’t forget to reverse the inequality

sign when dividing by a negative!

Example 2 The sum of two consecutive integers is at least negative fifteen. What are the smallest values of consecutive integers that will make this true?

Let x = the first integer x + 1 = the next consecutive integer

x + (x + 1) ≥ -15 2x + 1 ≥ -15

- 1 - 1 2x ≥ -16

2 2 x ≥ -8

x + 1 ≥ -7

The smallest values of consecutive integers

that will add to at least -15 are -8 and -7.

Example 3Connor went to the carnival with $22.50. He bought a hot dog and a drink for $3.75, and he wanted to spend the rest of his money on ride tickets which cost $1.25 each. What is the maximum number of ride tickets that he can buy?

Let r = the number of ride tickets he can buy

cost of food + cost of rides ≤ $22.50

3.75 + 1.25r ≤ 22.50 -3.75 -3.75 1.25r ≤ 18.75 1.25 1.25 r ≤ 15 tickets

Connor can buy a maximum of 15 ride tickets.

Example 4Stan earned $7.55 per hour plus an additional $100 in tips waiting tables on Saturday evening. He earned $160 in all. To the nearest hour, what is the least number of hours Stan would have to work to earn this much money?

Let h = the number of hours Stan will have to work

tips + hourly wages ≥ $160

100 + 7.55h ≥ 160 -100 -100 7.55h ≥ 60 7.55 7.55 x ≥ 7.9 hours

To the nearest hour, Stan would have to work at least 8 hours to earn $160.

Example 5Brenda has $500 in her bank account. Every week, she withdraws $40 for expenses. Without making any deposits, how many weeks can she withdraw this money if she wants to maintain a balance of at least $200?

Let w = the number of weeks Brenda withdraws money

starting account balance – money withdrawn ≥ $200

500 – 40w ≥ 200 -500 -500 – 40w ≥ -300

– 40 – 40 w ≤ 7.5 weeks

Don’t forget to reverse the inequality sign when dividing by a negative!

Brenda can withdraw $40 from the account for 7 full weeks and still have at least $200 in the account.

Example 3

x < 2x – 15

3

3x < 2x – 15

-2x -2x

x < -15

(3)(3)

TOO

1. -3 > -3t + 6

2. 26p – 20 > 14p + 64

3. .

1. t > 3

2. p > 7

3. t < -19

38

2

t

Special Case #1

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

3x + 4 > 2x + 6 + x

3x + 4 > 3x + 6

-3x -3x

4 > 6

• Distribute

•Combine like terms

•Get variable on same side

•True or false?

This state is FALSE:

No Solution

Special Case #2

3 – 3(y – 2) < 13 – 3(y – 6)

3 – 3y + 6 < 13 – 3y + 18

-3y + 9 < -3y + 31

+3y + 3y

9 < 31

• Distribute

•Combine like terms

•Get variable on same side

•True or false?

This state is TRUE:

All Numbers

Solve each inequality.

3 3 6t 3t

23 7

3w 15w

8 2 10 20n n 9n

Solve each inequality.

5 4 8 2 17m m 15m

2 15

3

xx

15x

6 2 4 12p p 7p

Solve each inequality.

3( 2) 8 44x x 10x

5( 4) 3( 4)k k 1k