Algebra Demystified 2nd Ed

312 algebra De mystif ieD

Inequalities and the Number Line

The solution to an algebraic inequality usually consists of a range (or ranges) of

numbers, which we write as an inequality in the form x < a, x ≤ a, x > a, or x ≥ a,

where a is a number. The inequality x < a means all numbers smaller than a but

not including a, and the inequality x ≤ a means all numbers smaller than a

including a itself. For example, the inequality x≤ 2 means all numbers smaller

than 2, including 2 itself. The inequality x < 2 means all numbers smaller than

2 but does not include 2. Similarly the inequality x > a means all numbers larger

than a but not a itself, and x ≥ a means all numbers larger than a including

a itself.

Every interval on the number line is represented by an inequality, and every

inequality is represented by an interval on the number line. We begin by repre-

senting inequalities with shaded regions on the number line. Later we will

represent inequalities with intervals.

The inequality x < a is represented on the number line by shading to the left

of the number a with an open dot at a.

a

FIgure 9-1

A closed dot at a is used for x < a.

FIgure 9-2

a

We shade to the right of a for x > a.

FIgure 9-3

a

Use a closed dot at a for x > a.

FIgure 9-4

a