3. Inequalities

inequality

A mathematical statement that compares two expressions by using one of the following signs: <, >, ≤, ≥, or ≠. "No more than" or "at most" means "less than or equal to." (≤) "At least" means "greater than or equal to." (₂)

solution of an inequality in one variable

A value or values that make the inequality true. Inequality: <color red>x + 2 < 6</color> Solution: <color red>x < 4</color>

absolute-value inequality

An inequality that contains a variable within an absolute value. The inequality <color red>IxI < a (when a > 0)<color red> asks, "What values of x have an absolute value less than <i>a</i>?" The solutions are numbers between <i>−a</i> and <i>a</i>. <color red>Ix + 4I > 7 x + 4 < 7 OR x + 4 > 7 x < 3 OR x > 3</color> Solution set: <color red>{x: 3 < OR x < 3}</color>

equivalent inequalities

Inequalities that have the same solution set. <color red>x > 5</color> and <color red>x + 2 > 7</color> are equivalent inequalities.

intersection

The intersection of two sets is the set of all elements that are common to both sets, denoted by <b>∩</b>. <color red>A = {1, 2, 3, 4} B = {1, 3, 5, 7, 9} A ∩ B = {1, 3}</color>

union

The union of two sets is the set of all elements that are in either set, denoted by <b>∪</b>.

compound inequality

Two inequalities that are combined into one statement by the word <i>and</i> or <i>or</i>. <color red>x ≥ 2 AND x < 7</color> (also written <color red>2 ≤ x < 7</color>) <color red>x < 2 OR x > 6</color>