## Solve and Graph Inequalities

## Introduction

Drawing the graph of a simple inequality such as y ≤ 15 can be done in three steps. First, draw a number line. Second, place the appropriate dot on the critical point(s) on the number line. Third, shade to show the solutions of the inequality. The graph of y ≤ 15 is shown below.

Some inequalities require manipulation before being graphed. Consider the inequality 3x – 2 < 10. While it may be possible to guess which solutions work for the inequality, trial and error is not the recommended way to solve it. Instead, use the rules of Algebra to simplify the inequality before attempting to graph it.

## Lesson

The rules of Algebra apply the same way to inequalities as they to do equations. The inequality can be manipulated by performing the same operation on both sides. Basically, the inequality 3x – 2 < 10 can be solved in the same way as the equation 3x – 2 = 10.

For more details on how to do the individual steps of the equation, see the lesson on solving two step equations. Example 1 shows how to solve a similar inequality.

**Example 1: **Simplify the inequality 8x + 4 ≤ 28

The inequality can be simplified using the rules of algebra.

The graph of example 1 shows that any number that is three or less is a solution to the original inequality. Any number greater than three does not work. To check your answer, try a few points and see if they work.

The points x = 2 and x = -1 are solutions to the inequality as expected from the graph. Likewise, doing the math for the point x = 6 demonstrates that it is not a solution to the inequality.

**Flipping the Inequality**

One unique aspect of simplifying inequalities is that they must sometimes be flipped during the simplification process. Flipping is necessary when multiplying or dividing the inequality by a negative number. Example 2 shows a problem where it is necessary to flip the inequality.

**Example 2****:** Graph the solutions to the inequality -m ≤ 6.

Think of flipping the inequality as a mirror image of the original. Suppose a number is greater than 5. The mirror image of that number would go past zero into the negatives and actually end up being less than negative five.

**Example 3****:** Graph the solutions to the inequality 12 – 3r > 30.

Mathematically, flipping the inequality is done only when multiplying or dividing by a negative. Another way to look at it is if the variable term starts out as a negative, then it will need to be multiplied or divided by a negative in order to isolate the variable. A negative variable term generally leads to flipping the inequality sign.

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