Algebra 1 · Algebra 2 · Grades 9, 10

How to Solve Absolute Value Equations

Quick answer

Absolute value measures distance from zero, so an equation like the absolute value of x equals 5 has two answers, 5 and -5. To solve, isolate the absolute value bars first, then split into two equations: one where the inside equals the positive value and one where it equals the negative. An absolute value can never equal a negative number, so those equations have no solution.

What you'll learn

  • Solve an absolute value equation by splitting it into two cases
  • Explain why two solutions appear, using distance from zero
  • Recognise when an absolute value equation has no solution

What absolute value means

The absolute value of a number is its distance from zero on the number line:

∣5∣=5∣−5∣=5∣0∣=0|5| = 5 \qquad |-5| = 5 \qquad |0| = 0

Distance has no direction, so it is never negative. Both 55 and −5-5 sit five units from zero, and absolute value reports that distance without saying which side.

Definition

∣x∣|x| is read “the absolute value of xx” and means the distance from xx to 00.

Why two answers appear

An equation like

∣x∣=5|x| = 5

asks: which numbers sit five units from zero? Walk five units right and you land on 55. Walk five units left and you land on −5-5. Two directions, two answers.

This is the whole reason the method splits into cases. You are not applying a rule about bars; you are answering a question that has two geometric answers.

The same question with a bigger expression inside works identically. In

∣x−3∣=4|x - 3| = 4

the quantity x−3x - 3 must sit four units from zero, so x−3=4x - 3 = 4 or x−3=−4x - 3 = -4.

And this is why a negative right-hand side is impossible:

∣x∣=−3|x| = -3

There is no point three units from zero on the negative side of the distance scale — distance is not a signed quantity. No number works, so there is no solution.

How to solve

  1. Isolate the absolute value so the bars stand alone on one side.
  2. If the other side is negative, stop: there is no solution.
  3. Split into two equations: inside == value, and inside =−= - value.
  4. Solve each.
  5. Check both answers in the original equation.

Worked examples

A note on absolute value inequalities

Once equations make sense, inequalities follow from the same distance idea, and each connects to a compound inequality:

FormAsksBecomes
∣x∣<5\lvert x \rvert < 5which points are within 5 of zero−5<x<5-5 < x < 5 (an AND)
∣x∣>5\lvert x \rvert > 5which points are further than 5 from zerox<−5x < -5 or x>5x > 5 (an OR)

“Less than” traps the value between two bounds; “greater than” splits it into two directions. Picture the number line and the right form follows without memorising.

Common mistakes

Practice problems

  1. Solve ∣x∣=9|x| = 9.

    Hint

    Which numbers sit nine units from zero?

    Answer

    x=9x = 9 or x=−9x = -9

    Full solution

    Both 99 and −9-9 are nine units from zero, so both satisfy the equation.

  2. Solve ∣x−5∣=2|x - 5| = 2.

    Answer

    x=7x = 7 or x=3x = 3

    Full solution

    Split: x−5=2x - 5 = 2 or x−5=−2x - 5 = -2, giving x=7x = 7 or x=3x = 3.

    Check: ∣7−5∣=2|7 - 5| = 2 ✓ and ∣3−5∣=∣−2∣=2|3 - 5| = |-2| = 2 ✓

  3. Solve ∣x+4∣=10|x + 4| = 10.

    Answer

    x=6x = 6 or x=−14x = -14

    Full solution

    Split: x+4=10x + 4 = 10 or x+4=−10x + 4 = -10, giving x=6x = 6 or x=−14x = -14.

    Check: ∣6+4∣=10|6 + 4| = 10 ✓ and ∣−14+4∣=∣−10∣=10|-14 + 4| = |-10| = 10 ✓

  4. Solve 3∣x∣=213|x| = 21.

    Hint

    Isolate the bars before splitting.

    Answer

    x=7x = 7 or x=−7x = -7

    Full solution

    Divide both sides by 33: ∣x∣=7|x| = 7. Split: x=7x = 7 or x=−7x = -7.

  5. Solve ∣x∣+6=2|x| + 6 = 2.

    Answer

    No solution.

    Full solution

    Subtract 66: ∣x∣=−4|x| = -4. A distance is never negative, so no value of xx works.

  6. Solve ∣2x−1∣=7|2x - 1| = 7.

    Answer

    x=4x = 4 or x=−3x = -3

    Full solution

    Split: 2x−1=72x - 1 = 7 gives 2x=82x = 8 and x=4x = 4; 2x−1=−72x - 1 = -7 gives 2x=−62x = -6 and x=−3x = -3.

    Check: ∣2(4)−1∣=7|2(4) - 1| = 7 ✓ and ∣2(−3)−1∣=∣−7∣=7|2(-3) - 1| = |-7| = 7 ✓

  7. Solve 4∣x+1∣−2=104|x + 1| - 2 = 10.

    Hint

    Two operations sit outside the bars. Undo both first.

    Answer

    x=2x = 2 or x=−4x = -4

    Full solution

    Add 22: 4∣x+1∣=124|x + 1| = 12. Divide by 44: ∣x+1∣=3|x + 1| = 3.

    Split: x+1=3x + 1 = 3 or x+1=−3x + 1 = -3, giving x=2x = 2 or x=−4x = -4.

    Check: 4∣2+1∣−2=12−2=104|2 + 1| - 2 = 12 - 2 = 10 ✓

  8. A machine cuts rods to a length of 50 cm with a tolerance of 0.4 cm. Write an absolute value equation for the two extreme lengths that are still acceptable, and find them.

    Hint

    The distance from the target length is exactly the tolerance at the extremes.

    Answer

    ∣L−50∣=0.4|L - 50| = 0.4, so L=50.4L = 50.4 cm or L=49.6L = 49.6 cm.

    Full solution

    Let LL be the length. At the extremes the distance from 5050 is exactly 0.40.4, so ∣L−50∣=0.4|L - 50| = 0.4.

    Split: L−50=0.4L - 50 = 0.4 gives L=50.4L = 50.4; L−50=−0.4L - 50 = -0.4 gives L=49.6L = 49.6.

    Any rod between these two lengths is inside tolerance, which would be written ∣L−50∣≤0.4|L - 50| \le 0.4.

Frequently asked questions

Why do absolute value equations usually have two answers?

Absolute value is distance from zero, and distance ignores direction. Two different numbers sit 5 units from zero, namely 5 and -5, so both satisfy the equation. You get two answers because two points are the same distance away.

Why must I isolate the bars before splitting?

Splitting relies on knowing the exact distance the inside is from zero. In 2 times the absolute value of x, plus 3, equals 11, the distance is not 11 — you must undo the times 2 and the plus 3 first to find that it is 4.

Why does an absolute value equal to a negative have no solution?

Distance is never negative. There is no point on the number line that sits -3 units from zero, so no value of x can make the equation true.

Key terms in this lesson

Absolute value
The absolute value of a number is its distance from zero on the number line. Distance ignores direction, so it is never negative: both 5 and -5 have absolute value 5.

Standards alignment

This lesson covers the following Common Core State Standards for Mathematics.

  • CCSS.MATH.CONTENT.HSA.REI.B.3Reasoning with Equations and InequalitiesSolve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
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