Inequalities and the Number Line

The inequality signs you must know are as follows: “greater than.” For example x 3 means “x is greater than three.” “less than.” For example x 8 means “x is less than eight.” “greater than or equal to.” For example x -2 means “x is greater than or equal to negative two.” “less than or equal to.” For example x 1 means “five is less than or equal to one.”

Plotting Inequalities on the Number Line The SAT and the ACT frequently ask students to graph inequalities on the number line as shown in the following examples. A closed (shaded) circle at the endpoint of the shaded portion of the number line indicates that the graph is inclusive of that endpoint, as in the case of or . An open (unshaded) circle at the endpoint of the shaded portion of the number line indicates that the graph is not inclusive of that endpoint, as in the case of or .

1) x

2) x

3) x -1

Sometimes you will be asked to plot multiple inequalities on a number line. Other times you will be asked to describe the inequality or inequalities indicated by the shaded part(s) of a number line. In problems that deal with multiple inequalities on one number line, it is crucial to understand the difference between and and or. And indicates that the both inequalities must be true for values in the solution set. This corresponds with one shaded region on the number line that is bounded on both sides. Two inequalities joined by and can also be written as a single statement. For instance, x and x can also be written as 3 x 5. Or indicates that only one of the inequalities must be true for a given value in the solution set. This corresponds with two shaded regions on the number line which typically go to infinity and negative infinity, respectively. Two inequalities joined by or cannot be written as a single statement as can be done with those joined by and.

4) x and x -2

Here, both conditions must apply because of the and, so the shaded region is the region that is less than 8 while at the same time greater than -2. Both endpoints are left unshaded because the inequality is noninclusive of the endpoints.

5) x or x

Here, because of the or, there are two separate shaded regions: one representing the values that are greater than 5, and the other representing the numbers that are less than or equal to 1. The 1 is shaded and the 5 is unshaded because the inequality is inclusive of the 1 but not the 5.

6) -3 x

Because these two inequalities are written as a single statement, it is an and situation, and is therefore represented by the single bounded interval in which both inequalities are true. The easiest way of reading a statement like this is “x is in between -3 and 4 (inclusive).” Both endpoints are shaded because the inequality is inclusive of both endpoints.

7) x or x

Here, because of the or, there are two separate shaded regions: one representing the values that are greater than -1, and the other representing the numbers that are less than -7. Both endpoints are left unshaded because the inequality is noninclusive of either endpoint.

Solving Inequalities Algebraically Sometimes, inequalities must be solved algebraically for a variable. Solve inequalities just like you solve normal equations. The only other thing you must remember is to flip the inequality sign when multiplying or dividing both sides of an equation by a negative number. A good justification for why you must do this is that 2 , but -2 -1.

8) 3x 18 x 6

9) x + 8 x -20

10) 2x + 5 17 2x 12 x 6

11) -4x 20 x -5 (Remember to flip the inequality sign when multiplying or dividing both sides by a negative number.)

12) -x – 3 5x + 15 -6x 18 x -3 (Remember to flip the inequality sign when multiplying or dividing both sides by a negative number.)