Linear Inequalities and Absolute Value

Grade 10 · Math · Study Guide

Solving and graphing linear inequalities, and solving absolute value equations and inequalities.

A linear inequality looks like a linear equation but uses <, >, ≤, or ≥ instead of =. Solving it is almost identical to solving an equation — with one critical difference: when you multiply or divide both sides by a negative number, you must flip the inequality sign. Forgetting this rule is the most common mistake in this topic.

The solution to a linear inequality in one variable is a range of values, graphed on a number line. Use an open circle for strict inequalities (< or >) and a closed circle for inequalities that include the endpoint (≤ or ≥). For compound inequalities, 'and' inequalities (like -3 < x < 5) represent an intersection (both must be true), while 'or' inequalities represent a union (either can be true).

Absolute value represents distance from zero on the number line — it's always non-negative. |x| = 5 means x is 5 units from zero, giving two solutions: x = 5 or x = -5. For |ax + b| = c (where c ≥ 0), split into two cases: ax + b = c and ax + b = -c, then solve each.

Absolute value inequalities split into two cases too, but the direction depends on the type: |x| < c means -c < x < c (between; an 'and' compound), while |x| > c means x < -c or x > c (outside; an 'or' compound). A helpful memory trick: less than means 'less-AND,' greater than means 'greatOR.'

Key terms

Linear Inequality: A statement using <, >, ≤, or ≥ comparing a linear expression to a value.
Flip the Sign: When multiplying or dividing an inequality by a negative number, the inequality direction reverses.
Compound Inequality: Two inequalities joined by 'and' (intersection) or 'or' (union).
Absolute Value: |x| represents the distance from zero; always ≥ 0.
Absolute Value Inequality: |x| < c gives -c < x < c; |x| > c gives x < -c or x > c.

Flashcards

Inequality flip rule: Multiply or divide by a negative → flip the inequality sign
|x| = c solution: x = c or x = -c (two solutions when c > 0)
|x| < c solution: -c < x < c (between; 'and' compound inequality)
|x| > c solution: x < -c or x > c (outside; 'or' compound inequality)
Open circle on number line: Strict inequality (< or >); endpoint NOT included
Closed circle on number line: Non-strict inequality (≤ or ≥); endpoint IS included

Practice questions

Solve: -3x > 12. What is x?
1. x > -4
2. x < -4
3. x > 4
4. x < 4
Show answer
Answer: x < -4
Divide both sides by -3 AND flip the sign: -3x > 12 → x < -4.
When dividing both sides of an inequality by a negative number, the inequality sign must be flipped.
Show answer
Answer: True
Multiplying or dividing by a negative reverses the order relationship — the inequality sign must flip.
|x - 3| = 7 gives x = 10 or x = ___.
Show answer
Answer: -4
Split: x-3=7 → x=10; and x-3=-7 → x=-4. Both solutions are valid.
The solution to |x| < 4 is:
1. x > 4 or x < -4
2. -4 < x < 4
3. x > -4
4. x < 4
Show answer
Answer: -4 < x < 4
|x| < 4 means x is within 4 units of zero: -4 < x < 4.
An open circle on a number line graph represents that the endpoint is included in the solution.
Show answer
Answer: False
An open circle means the endpoint is NOT included (strict inequality). A closed (filled) circle means it IS included.

Fun facts

Temperature 'within 5 degrees of 70°F' is a real-world absolute value inequality: |T - 70| ≤ 5, which means 65°F ≤ T ≤ 75°F.
The concept of absolute value extends to complex numbers (where it's called the modulus) and to vectors (where it becomes magnitude) — it's a universal measure of 'size' across all of mathematics.

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