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### Key Concepts

• Write and solve multi-step inequalities

• Solve more multi-step inequalities

• Solve multi—step inequalities by combining like terms

**5.7 Solve Multi-Step Inequalities **

Solving multi-step inequalities is very similar to solving equations—what you do to one side, you need to do to the other side in order to maintain the “balance” of the inequality.

**Example:**

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

The solutions are all real numbers greater than 6.

**5.7.1. Write and Solve Multi-Step Inequalities **

**Multi-step: **Involving two or more distinct steps or stages.

**Example 1:**

Jose is starting a word-processing business out of his home. He plans to charge $15 per hour. He anticipates his monthly expenses to be $490 for equipment rental, $45 for materials, and phone usage.

Write and solve an inequality to find the number of hours he must work in a month for a profit of at least $600.

**Sol:**

Define a variable:

h =number of hours worked

Write an inequality to model the problem:

15h – (490 + 45 + 65) > 600

15h – (600) +600 > 600 + 600 Add 600 to both sides.

15h > 1200 Divide both sides by 15.

** h ****>**** 80**

Jose must work at least 80 hours.

**Example 2:**

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

**Sol:**

3 + 2(x+4) > 3 Distributive 2 on the left side.

3 + 2x + 8 > 3 Combine like terms.

2x + 11 > 3 Since 11 is added to 2x, subtract 11 from both sides to undo the addition.

2x > –8 Divide both sides by 2.

x > –4

Graph the solution.

**5.7.2. Solve More Multi-Step Inequalities **

**Example:**

Solve the inequality –4(2 – x) < 8. Then graph the solution.

**Sol:**

–4(2 – x) < 8 Distributive -4 on the left side.

–4(2) – 4(–x) < 8

–8 +4x < 8 Since -8 is added to 4x, add 8 to both sides.

–8 + 8 + 4x < 8 + 8

4x < 16 Since *x* is multiplied by 4, divide both sides by 4 to undo the multiplication.

**X ****<**** 4**

Graph the solution.

**5.7.3. Solve Multi—Step Inequalities by Combining Like Terms**

**Like terms:**

Like terms are terms that have the same variables and powers. The coefficients do not need to match.

**Example:**

Edith is counting the number of seeds in her fruit. Her pomegranate has one less than three times as many seeds as her apple. Her orange has thirteen less than five times as many seeds as her apple. If her pomegranate has more seeds than her orange, how many seeds are there in Edith’s apple?

**Sol:**

*n* represents the number of seeds in Edith’s apple.

Pom seeds Orange seeds

3n – 1 > 5n – 13

3n > 5n – 12

–2n > –12

n < 6 apple seeds

## Exercise

- Solve the inequality -2(x + 3) +2 Z 6. Then graph the solution.

2. Solve the inequality -1 – 6(6 + 2x) < 11. Then grapl the solution.

3. Solve the inequality 18 < -3(4x – 2).Then graph the solution.

4. The length of a picture frame is 7 inches more than the width. For what values of xis the perimeter of the picture frame greater than 154 inches?

5. Solve the inequality 5(2t + 3) -3t < 16.

6. Solve the inequality 15.6 <2.7 ((z – 1) – 0.6.

7. Solve 2(3y – 5) < -16.

8. – 4(6n + 7) 122

9. -9(q + 3) < 45

10.5x + 2(x + 1) 23

### Concept Map

### What have we learned:

• Solve multi- step inequalities

• Write and solve multi-step inequalities

• Solve more multi-step inequalities

• Identify like terms

• Solve multi-step inequalities by combining like terms

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