Inequalities are similar to equations, except both sides of the problem are not necessarily equal (≠). Inequalities may be represented as follows:

greater than (>)

greater than or equal to (≥)

less than (<)

less than or equal to (≤)

Solving Inequalities

Inequalities can be solved by manipulating, just like equations. The only difference is that the direction of the inequality sign must be reversed when the inequality is divided by a negative number.

10 – 2x > 40

– 2x > 4

x < –2

The solution to an inequality is a set of numbers, not a single value. For example, simplifying 4x + 2 ≤ 14 gives the inequality x ≤ 3, meaning every number less than or equal to 3 would be included in the set of correct answers.

Compound Inequalities

Compound inequalities have more than one inequality expression.

5 < x < 12 → x > 5 and x < 12

Inequalities joined by and are intersections. The solution to these compound inequalities will be all the values that make both inequalities true.

Inequalities joined by or are unions. The solution to a union will be all the values that make either inequality true.

Solve the inequality:

–1 ≤ 3(x + 2) – 1 ≤ x + 3

–1 ≤ 3(x + 2) – 1 3(x + 2) – 1 ≤ x + 3

x ≤ –1 –2 ≤ x

–2 ≤ x ≤ –1

Graphing Linear Inequalities

Inequalities with one variable may be represented on a number line, as shown below. A circle is placed on the end point with a filled circle representing and and an empty circle representing < and >. An arrow is then drawn to show either all the values greater than or less than the value circled.

Linear inequalities in two variables can be graphed the same way as linear equations. Start by graphing the corresponding equation of a line (temporarily replace the inequality with an equal sign, and then graph). If the inequality is a “greater/less than,” a dashed line is used; a solid line is used to indicate “greater/less than or equal to.”

A dashed line is used for “greater/less than” because the solution may approach that line, but the coordinates on the line can never be a solution.

One side of the boundary line is the set of all points (x, y) that make the inequality true. This side is shaded to indicate that all these values are solutions.

The simplest method to determine which side should be shaded is to choose a point (x, y) on one side of the boundary and evaluate the inequality, substituting these x- and y-values. If the point makes the inequality true, that side is shaded; if it does not, it is not a solution, so the other side is shaded.