### 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

#### Related topics

#### Addition and Multiplication Using Counters & Bar-Diagrams

Introduction: We can find the solution to the word problem by solving it. Here, in this topic, we can use 3 methods to find the solution. 1. Add using counters 2. Use factors to get the product 3. Write equations to find the unknown. Addition Equation: 8+8+8 =? Multiplication equation: 3×8=? Example 1: Andrew has […]

Read More >>#### Dilation: Definitions, Characteristics, and Similarities

Understanding Dilation A dilation is a transformation that produces an image that is of the same shape and different sizes. Dilation that creates a larger image is called enlargement. Describing Dilation Dilation of Scale Factor 2 The following figure undergoes a dilation with a scale factor of 2 giving an image A’ (2, 4), B’ […]

Read More >>#### How to Write and Interpret Numerical Expressions?

Write numerical expressions What is the Meaning of Numerical Expression? A numerical expression is a combination of numbers and integers using basic operations such as addition, subtraction, multiplication, or division. The word PEMDAS stands for: P → Parentheses E → Exponents M → Multiplication D → Division A → Addition S → Subtraction Some examples […]

Read More >>#### System of Linear Inequalities and Equations

Introduction: Systems of Linear Inequalities: A system of linear inequalities is a set of two or more linear inequalities in the same variables. The following example illustrates this, y < x + 2…………..Inequality 1 y ≥ 2x − 1…………Inequality 2 Solution of a System of Linear Inequalities: A solution of a system of linear inequalities […]

Read More >>
Comments: