HexMathv1.0.3

Solve absolute value equations with steps

Type or scan an absolute value equation. HexMath splits it into cases and checks each answer.

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What this page covers

  • Equations with one absolute value, after isolating it on one side.
  • An absolute value equal to an expression in x, where each candidate must be checked.
  • Absolute values on both sides of the equation.
  • Not covered here: absolute value inequalities such as |x − 2| ≤ 5 (use Inequality calculator with steps).

How to enter the problem

  • Type a vertical bar | on each side of the expression, such as |2x - 3| = 7.
  • Scan it. Fit the one problem in the frame.
  • Choose a photo. Several problems in one photo are solved as a list.
  • Then tap Solve.

Worked examples

Example 1

Problem

Solve 3∣x+2∣−4=11\displaystyle 3|x + 2| - 4 = 11.

Answer

Verified

The answer is

x=3,x=−7x = 3, x = -7

Explanation

  1. Isolate the absolute value
    Add 44, then divide by 33
    ∣x+2∣=5|x + 2| = 5
  2. Split into two cases
    x+2=5x + 2 = 5 or x+2=−5x + 2 = -5.
  3. Solve each case
    x=3x = 3 or x=−7x = -7.

Example 2

Problem

Solve ∣x+4∣=3x\displaystyle |x + 4| = 3x.

Answer

Verified

The answer is

x=2x = 2

Explanation

  1. Note the condition
    An absolute value is never negative, so 3x≥03x \geq 0, that is x≥0x \geq 0.
  2. Positive case
    x+4=3xx + 4 = 3x gives x=2x = 2.
  3. Negative case
    x+4=−3xx + 4 = -3x gives x=−1x = -1.
  4. Check each candidate
    x=2x = 2: ∣6∣=6|6| = 6, true. x=−1x = -1: ∣3∣=3|3| = 3 but 3x=−33x = -3, false, so −1-1 is rejected.

Example 3

Problem

Solve ∣x−1∣=∣2x+3∣\displaystyle |x - 1| = |2x + 3|.

Answer

Verified

The answer is

x=−4,x=−23x = -4, x = -\frac{2}{3}

Explanation

  1. Use equal distances
    Two absolute values are equal when the insides are equal or opposite.
  2. Equal case
    x−1=2x+3x - 1 = 2x + 3 gives x=−4x = -4.
  3. Opposite case
    x−1=−(2x+3)x - 1 = -(2x + 3) gives 3x=−23x = -2, so x=−23x = -\frac{2}{3}.
  4. Check
    x=−4x = -4: ∣−5∣=∣−5∣|-5| = |-5|. x=−23x = -\frac{2}{3}: ∣−53∣=∣53∣|-\frac{5}{3}| = |\frac{5}{3}|. Both hold.

Common mistakes

  • Solving only the positive case. ∣2x−3∣=7|2x - 3| = 7 has two solutions, one from 2x−3=72x - 3 = 7 and one from 2x−3=−72x - 3 = -7.
  • Splitting into cases before isolating. In 3∣x+2∣−4=113|x + 2| - 4 = 11, first reach ∣x+2∣=5|x + 2| = 5.
  • Keeping a candidate that fails the check, such as x=−1x = -1 in ∣x+4∣=3x|x + 4| = 3x, where the left side is 33 and the right side is −3-3.
  • Splitting when the other side is negative. ∣2x−3∣=−7|2x - 3| = -7 has no solution, since an absolute value is never negative.

Checks, assumptions and limits

  • Substitute each answer. For the runnable example, x=5x = 5 gives ∣7∣=7|7| = 7 and x=−2x = -2 gives ∣−7∣=7|-7| = 7.
  • An absolute value is a distance from 0, so it is never negative.
  • When the other side contains x, every candidate must be checked in the original equation.
  • HexMath can make mistakes. Double check important steps.

Related tasks

Frequently asked questions

Two numbers sit at the same distance from 0. If |A| = 7, then A is 7 or -7, and each case gives its own solution.

Then there is no solution. An absolute value is a distance, so it can be 0 or positive but never negative.

Yes. Use the camera on the one equation, or choose a photo. Several problems in one photo are solved as a list.

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