### Key Concepts

• Graph the solutions of an inequality.

• Graph to solve an inequality.

• Substitute to solve an inequality.

**Introduction:**

## Equality vs inequality:

An equation shows when expressions are equal. Equations use equal signs (=).

An inequality is a statement that uses the greater-than symbol (>), the less-than symbol (<), the greater-than-or-equal-to symbol (≥), or the less-than-or-equal-to symbol (≤).

**Variables in inequality:**

Variables can be used with inequalities. A variable in an inequality stands for all numbers that make the inequality true.

For example, in the inequality *x *< 4, the *x *stands for all numbers less than 4. So *x *can be 0, 1, 2 or 3.

The inequality 12 ≤ *y *+ 5 can have solutions *y *= 7, 8, and 9, since 7 + 5 = 12, 8 + 5 = 13, and 9 + 5 = 14.

**4.7.1 Graph the solutions of an inequality**

**Example 1:**

Graph all the solutions of *x* < 3.

**Solution:**

**Step 1:**

To graph *x *< 3, draw an open circle at 3 on a number line.

**Step 2:**

Find some solutions and plot them on a number line.

**Step 3:**

Start at the open circle and shade the solutions you found.

**4.7.2 Graph to solve an inequality**

**Example 2:**

The temperature in a greenhouse should be 57 degrees or higher. Write an inequality to describe the allowable temperature in the greenhouse.

**Solution:**

Write and graph an inequality.

The possible temperatures in the greenhouse, *t*, are greater than or equal to 57 degrees.

**Example 3:**

The maximum weight allowed by law in a freight elevator is 1,500 pounds. Let *w* represent the weight on the elevator. Write an inequality to describe the allowable weight on the elevator.

**Solution:**

Write and graph an inequality.

The possible weights in the elevator, *w*, are less than or equal to 1,500 pounds.

**4.7.3 Substitute to solve an inequality**

**Example 4:**

John’s family is planning to go on vacation. They are planning to visit any one place which is beyond 250 miles from their home town. Which place, if any, can be chosen for the family vacation?

**Solution:**

Write an inequality to represent the situation.

## Exercise:

1. Name 3 solutions for z> S.

2. Name 3 solutions for x 4.

3. How many solutions does the inequality x > 18 have? Explain.

4. Write the inequality the following graph represents.

5. Write the inequality the following graph represents.

6. Substitute each given value of the variable to find which, if any, is a solution of the inequality.

x < 8 x = 4.2, 5.2, 8.2, 9

7. Substitute each given value of the variable to find which, if any, is a solution of the inequality.

s > 25 s = 24,25, 25.1, 27

8. Substitute each given value of the variable to find which, if any, is a solution of the inequality.

t < 4 t = 0, 4, 5,6

9. Name three solutions of the inequality t 24.

10. Tina started a graph to show the inequality y < 3.4. Finish labeling the number line and draw the graph.

### What have we learned:

• Graph the solutions of an inequality on a number line.

• Graph to solve an inequality on a number line.

• Substitute a set of values in a variable to solve an inequality.

### Concept Map

#### Related topics

#### Find Common Denominators

Key Concepts Introduction In this chapter, we will learn about common denominators, finding equivalent fractions and finding common denominators. Common denominator If two or more fractions have the same number as the denominator, then we can say that the fractions have a common denominator. Let us understand the common denominator in detail: In this pizza, […]

Read More >>#### Composite Figures – Area and Volume

A composite figure is made up of simple geometric shapes. It is a 2-dimensional figure of basic two-dimensional shapes such as squares, triangles, rectangles, circles, etc. There are various shapes whose areas are different from one another. Everything has an area they occupy, from the laptop to your book. To understand the dynamics of composite […]

Read More >>#### Special Right Triangles: Types, Formulas, with Solved Examples.

Learn all about special right triangles- their types, formulas, and examples explained in detail for a better understanding. What are the shortcut ratios for the side lengths of special right triangles 30 60 90 and 45 45 90? How are these ratios related to the Pythagorean theorem? Right Angle Triangles A triangle with a ninety-degree […]

Read More >>#### Ways to Simplify Algebraic Expressions

Simplify algebraic expressions in Mathematics is a collection of various numeric expressions that multiple philosophers and historians have brought down. Talking of algebra, this branch of mathematics deals with the oldest concepts of mathematical sciences, geometry, and number theory. It is one of the earliest branches in the history of mathematics. The study of mathematical […]

Read More >>
Comments: