Inequalities and Absolute Value Equations

Grade 9 · Math · Study Guide

Solve and graph linear inequalities and absolute value equations and inequalities on the number line.

An inequality is a mathematical statement comparing two expressions using <, >, ≤, or ≥. Solving a linear inequality is similar to solving an equation, with one critical rule: when you multiply or divide both sides by a negative number, you must flip (reverse) the inequality sign.

For example, −2x > 8 becomes x < −4 after dividing both sides by −2. Solutions to inequalities are graphed on a number line with open circles for strict inequalities (< and >) and closed circles for inclusive inequalities (≤ and ≥).

Compound inequalities combine two conditions. An AND compound inequality like 2 < x ≤ 7 means x must satisfy both conditions simultaneously (intersection). An OR compound inequality like x < −1 or x > 4 means x satisfies at least one condition (union).

The absolute value |x| represents the distance of x from zero on the number line, so |x| is always non-negative. For |x| = a (where a > 0), the solutions are x = a or x = −a. For |x| < a, the solution is −a < x < a. For |x| > a, the solution is x < −a or x > a. Always isolate the absolute value before splitting into cases.

Key terms

Linear inequality
A mathematical statement using <, >, ≤, or ≥ that compares linear expressions
Inequality reversal rule
When multiplying or dividing both sides of an inequality by a negative number, the inequality sign must be flipped
Compound inequality (AND)
Two inequalities joined by 'and'; the solution is the intersection of both solution sets
Compound inequality (OR)
Two inequalities joined by 'or'; the solution is the union of both solution sets
Absolute value
The distance of a number from zero on a number line, always non-negative; written |x|
Absolute value equation
An equation of the form |expression| = a, which typically has two solutions

Flashcards

Solve −5x > 25
x < −5 (flip sign when dividing by −5)
Graph of x ≤ 3
Closed circle at 3, shaded to the left
Solve |x| = 6
x = 6 or x = −6
Solve |x + 2| > 5
x > 3 or x < −7 (split into two cases, both going outward)
Solve |3x − 6| = 0
x = 2 (only one solution when the absolute value equals zero)
AND compound inequality
3 < 2x − 1 < 9 → 4 < 2x < 10 → 2 < x < 5
When does |expression| = a have no solution?
When a < 0, because absolute value is never negative
Solve 2|x| − 3 < 7
2|x| < 10 → |x| < 5 → −5 < x < 5

Questions

  1. Solve: −3x + 6 > 15

    1. x > −3
    2. x < −3
    3. x > 3
    4. x < 3
    Show answer

    Answer: x < −3

    −3x + 6 > 15 → −3x > 9 → x < −3 (flip sign when dividing by −3).

  2. Solve |x − 4| = 7. The two solutions are x = 11 and x = __.

    Show answer

    Answer: -3

    |x − 4| = 7 means x − 4 = 7 (giving x = 11) or x − 4 = −7 (giving x = −3).

  3. Solve |2x + 1| < 9.

    1. −5 < x < 4
    2. x < −5 or x > 4
    3. −4 < x < 5
    4. x < −4 or x > 5
    Show answer

    Answer: −5 < x < 4

    |2x + 1| < 9 means −9 < 2x + 1 < 9. Subtract 1: −10 < 2x < 8. Divide by 2: −5 < x < 4.

  4. The solution to −4x ≥ 20 is x ≥ −5.

    Show answer

    Answer: False

    −4x ≥ 20. Divide by −4 and flip: x ≤ −5. The sign flips from ≥ to ≤.

  5. Which graph represents the solution to −2 ≤ x < 5?

    1. Open circle at −2, closed circle at 5
    2. Closed circle at −2, open circle at 5
    3. Two open circles at −2 and 5
    4. Two closed circles at −2 and 5
    Show answer

    Answer: Closed circle at −2, open circle at 5

    −2 ≤ x uses ≤ (closed/filled circle at −2). x < 5 uses < (open circle at 5). Shaded between them.

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