Absolute Value: Rules, Equations & Inequalities

Q: How do I solve absolute value equations?

Split into two cases: |x − 3| = 7 means x − 3 = 7 OR x − 3 = −7 → x = 10 or x = −4. Steps: isolate the absolute value FIRST (2|x| + 3 = 9 → |x| = 3), then split. If the right side is negative (|x| = −3), no solution — distances cannot be negative. Always check both answers in the original equation.

Q: How do absolute value inequalities work?

Two patterns to memorize: |A| b becomes the OR pair A > b or A < −b (far from zero). |x − 2| < 3 → −3 < x − 2 < 3 → −1 < x < 5. |x + 1| ≥ 4 → x ≥ 3 or x ≤ −5. "Less than" wraps, "greater than" splits — draw the number line once and it sticks forever.

Q: What does the absolute value graph look like?

A V shape with its vertex at the origin: straight line y = x on the right, y = −x on the left, corner at (0,0). |x − 3| + 2 shifts the V right 3 and up 2 (vertex at (3,2)). The corner point is non-differentiable — the slope jumps from −1 to +1. Recognizing the V instantly explains why absolute equations produce two symmetric solutions.

Q: Where is absolute value used in practice?

Q: What mistakes happen with absolute value?

(1) |x − 3| = 7 → x − 3 = 7 only (forgetting the −7 case); (2) splitting before isolating: 2|x| + 3 = 9 does NOT give 2x + 3 = ±9; (3) |x| b is OR; (5) thinking |a + b| = |a| + |b| always — only same-sign numbers; |(−3) + 5| = 2 ≠ 8. The triangle inequality gives ≤, not =.

Key Takeaways

Definition: |x| = x if x ≥ 0, −x if x < 0 — the distance from zero, never negative

Equations split: |A| = b → A = b OR A = −b (b ≥ 0 required for solutions)

Inequalities: |A| < b → −b < A < b (AND); |A| > b → A > b OR A < −b

Key properties: |ab| = |a||b|; |a+b| ≤ |a|+|b| (triangle inequality)

Core answer: |x| is the distance from zero, always non-negative: |5| = 5, |−5| = 5, |0| = 0. To solve |x − 3| = 7, split into two cases: x − 3 = 7 or x − 3 = −7, giving x = 10 or x = −4. For inequalities, |x| < a means −a < x < a (an interval); |x| > a means x < −a or x > a (two rays).

The three ways to read |x − a|

Distance: |x − 3| is the distance between x and 3 on the number line.
Piecewise: |x − 3| = x − 3 when x ≥ 3, else 3 − x.
Tolerance: |x − 10| ≤ 0.5 means "within 0.5 of 10" — the language of engineering tolerances and error bounds.

Worked examples

Example 1 — Equation. |2x − 1| = 5 → 2x − 1 = 5 or 2x − 1 = −5 → x = 3 or x = −2. Check both: |5| = 5 ✓, |−5| = 5 ✓.

Example 2 — Tolerance. A resistor marked 1 kΩ ±5%: actual value R satisfies |R − 1000| ≤ 50, i.e., 950–1050 Ω.

Example 3 — Temperature control. A vaccine fridge must hold |T − 4| ≤ 2 °C: the safe band is 2–6 °C; alarms fire outside it.

Example 4 — |x − 2| + |x − 8| = 6. Distances to 2 and 8 summing to their separation means x lies anywhere between them: every x in [2, 8] is a solution.

Common mistakes and myths

"|x| = x" — only for x ≥ 0; for negatives |x| = −x (which is positive).
Squaring without care — |a|² = a² always, so squaring both sides of |x−1| = |2x+3| is legal and avoids case-splitting: (x−1)² = (2x+3)².
Wrong inequality direction — |x| < 3 is a band (−3, 3); |x| > 3 is outside it. Memorize: "less is between, greater is beyond".
Forgetting the empty case — |x − 5| = −2 has no solution; absolute values never equal negatives.
Assuming |a + b| = |a| + |b| — only when signs agree; |3 − 5| = 2 but |3| + |−5| = 8. The triangle inequality says ≤, not =.

Frequently Asked Questions

What is absolute value?

The distance from zero on the number line — always non-negative. |7| = 7, |−7| = 7, |0| = 0. Formally: |x| = x when x ≥ 0, and −x when x < 0. It answers "how far?" not "which direction?" — temperature deviation, error margins, and distance all use absolute value because direction does not matter there.

How do I solve absolute value equations?

Split into two cases: |x − 3| = 7 means x − 3 = 7 OR x − 3 = −7 → x = 10 or x = −4. Steps: isolate the absolute value FIRST (2|x| + 3 = 9 → |x| = 3), then split. If the right side is negative (|x| = −3), no solution — distances cannot be negative. Always check both answers in the original equation.

How do absolute value inequalities work?

Two patterns to memorize: |A| < b becomes the AND sandwich −b < A < b (close to zero). |A| > b becomes the OR pair A > b or A < −b (far from zero). |x − 2| < 3 → −3 < x − 2 < 3 → −1 < x < 5. |x + 1| ≥ 4 → x ≥ 3 or x ≤ −5. "Less than" wraps, "greater than" splits — draw the number line once and it sticks forever.

What does the absolute value graph look like?

A V shape with its vertex at the origin: straight line y = x on the right, y = −x on the left, corner at (0,0). |x − 3| + 2 shifts the V right 3 and up 2 (vertex at (3,2)). The corner point is non-differentiable — the slope jumps from −1 to +1. Recognizing the V instantly explains why absolute equations produce two symmetric solutions.

Where is absolute value used in practice?

What mistakes happen with absolute value?

(1) |x − 3| = 7 → x − 3 = 7 only (forgetting the −7 case); (2) splitting before isolating: 2|x| + 3 = 9 does NOT give 2x + 3 = ±9; (3) |x| < −1 "solutions" — impossible, empty set; (4) flipping the inequality wrongly: |A| < b is AND, |A| > b is OR; (5) thinking |a + b| = |a| + |b| always — only same-sign numbers; |(−3) + 5| = 2 ≠ 8. The triangle inequality gives ≤, not =.