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➕ Algebra 1 · Inequalities

Absolute Value Equations: notes and common mistakes

Study notes

Key ideas

  • |x| is the distance from x to 0 on the number line, so it is never negative. |x − c| is the distance from x to c.
  • For |A| = c, where A is any expression: if c > 0, then A = c or A = −c. If c = 0, then A = 0. If c < 0, there is no solution.
  • Get the absolute value alone before splitting. Undo the added or subtracted numbers, then divide by the coefficient (not zero), keeping the bars together as one unit.
  • Check both answers in the original equation.
  • A tolerance problem |x − T| = d gives the two edge values T − d and T + d. The values that pass are all the numbers between them, edges included: T − d ≤ x ≤ T + d.

Steps

  1. Isolate the absolute value: add or subtract the numbers around it, then divide both sides by its coefficient (not zero).
  2. Look at the number on the right. Negative: no solution, stop. Zero: one equation. Positive: two equations.
  3. Write inside = c and inside = −c.
  4. Solve each equation.
  5. Check each answer in the original equation.

Formulas

  • Two solutions: |A| = c with c > 0 gives A = c or A = −c
  • One solution: |A| = 0 gives A = 0
  • No solution: |A| = c with c < 0 has no solution
  • Distance form: |x − c| = d with d > 0 gives x = c − d or x = c + d

Watch out

  • Isolate the bars first: in 3|x + 4| − 6 = 15, do not split until you have |x + 4| = 7.
  • In the second equation, change the sign of the number on the right (not just one term inside the bars).
  • A negative number on the right, after the bars are alone, means no solution. Do not solve it anyway.
  • The center of |x − 3| is 3, and the center of |x + 2| is −2 (the value of x that makes the inside 0, when x has coefficient 1). So |x + 2| = 5 means 5 units from −2.
  • Check in the original equation, not in a line you rewrote.

Why it works

  • Why two solutions? Two different numbers can be the same distance from a center, one on each side. Why only one when the distance is 0? Only the center itself is 0 units from the center.
  • Why isolate first? The rule says what the distance equals. In 3|x + 4| − 6 = 15, the distance |x + 4| is not 15. It is 7, once you undo the −6 and the multiplication by 3.
  • Why does a negative number on the right give no solution? An absolute value is a distance, and a distance is never negative.

Common mistakes

✗ Mistake|x| = 5 means x = 5.
✓ InsteadBoth 5 and −5 are 5 units from 0, so x = 5 or x = −5. Forgetting the second case is a common slip.
✗ Mistake|x − 2| = 7 gives x = 9 or x = −9.
✓ InsteadThe second case is x − 2 = −7, which gives x = −5. The solutions are the two numbers 7 units from 2: 9 and −5. Check: |−5 − 2| = 7.
✗ MistakeIn 4|x + 7| = 20, write x + 7 = 20 or x + 7 = −20.
✓ InsteadIsolate first: divide by 4 to get |x + 7| = 5. Then x + 7 = 5 or x + 7 = −5, so x = −2 or x = −12. The wrong split gives x = 13, and 4|13 + 7| = 80, not 20.
✗ Mistake|x − 6| = −5 means x − 6 = −5, so x = 1.
✓ InsteadNo number has an absolute value of −5. Check: |1 − 6| = 5, not −5. There is no solution.
✗ Mistake|x − 4| is the same as |x| − 4, so you can take the bars off each piece.
✓ InsteadThe bars apply to the whole expression inside. Try x = 1: |1 − 4| = 3, but |1| − 4 = −3.

Teach it back

Explain to a friend how to solve 2|x − 6| + 1 = 9, and what it would mean if the bars ended up equal to a negative number.

Explaining it out loud shows what you really know. Did you…

  • I isolated the bars: subtract 1 to get 2|x − 6| = 8, then divide by 2 to get |x − 6| = 4.
  • I split into x − 6 = 4 or x − 6 = −4 and found x = 10 or x = 2.
  • I checked both in the original equation: 2|10 − 6| + 1 = 9 and 2|2 − 6| + 1 = 9.
  • I said that if the bars equal a negative number, there is no solution, because a distance is never negative.

Your 3-week memory plan

  1. TodayRead the lesson, then try the flashcards.
  2. TomorrowRecall it from memory first, then check.
  3. In 3 daysTake the true/false and practice questions.
  4. In 1 weekMix it with older topics and teach it back.
  5. In 3 weeksA last quick review before it fades.

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