REVIEW: Solving Linear Inequalities

Previous Material: Solving Linear Inequalities Part 1

READ: Recognize Equivalent Inequalities

Recognize Equivalent Inequalities

Sometimes, you may need to rewrite an inequality as an equivalent inequality in order to better understand it. This means that an inequality can be written in two different ways so that we can understand it. Look at the example below.

Example 1

Graph this inequality 4 \ge x.

Draw a number line from –5 to 5.

The inequality 4 \ge x is read as “4 is greater than or equal to x.” This inequality will be easier to understand if we rewrite it so that the variable is listed first. If we list the x first, we must reverse the inequality symbol. That means changing the “greater than or equal to” symbol (\ge) to a “less than or equal to symbol” \le.

4 \ge x is equivalent to x \le 4.

This makes sense. If 4 is greater than or equal to x, then x must be less than or equal to 4.

The inequality x \le 4 is read as “x is less than or equal to 4.” So, the solutions of this inequality include 4 and all numbers that are less than 4. Draw a closed circle at 4 to show that 4 is a solution for this inequality. Then draw an arrow showing all numbers less than 4.

The graph above shows the solutions for the inequality x \le 4.

Example 2

Write an equivalent inequality for x \ge 5.

To write an equivalent inequality, we reverse the terms. If x is greater than or equal to five then five is less than or equal to x. Let’s rewrite this.

The equivalent inequalities are x \ge 5and 5 \le x.