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Algebra Absolute value equations and inequalities

Absolute value equations

20 practice questions 2 video lessons Theory + worked examples
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Theory

Absolute value is distance from zero, so it is never negative. An equation \(|X|=a\) (with \(a\ge0\)) has two cases:
\[|X|=a\ \Longrightarrow\ X=a\ \text{or}\ X=-a.\]
Isolate the absolute value first, and if it equals a negative there is no solution.
Number line Absolute value equals 5 gives two points, 5 units from zero. -5 0 5 |x| = 5: two solutions, x = ±5
\(|x|=5\) gives two points at distance \(5\).
Absolute value equations Absolute value equations Absolute value equations |X| = a (a ≥ 0) → X = a or X = -a two cases: positive and negative |X| = negative → no solution
The two-case method.

The case split:

\[|X|=a\ \Rightarrow\ X=\pm a\quad(a\ge0)\]
absolute value of X equals a means X equals plus or minus a
\(|X|=\)negative has no solution.

How to solve

  1. Isolate the absolute value.
  2. If it equals a negative, state no solution.
  3. Otherwise write two equations with \(\pm\).
  4. Solve each and check.
Example 1 — Basic
Solve \(|x|=5\).
Solution

The inside is \(5\) or \(-5\).

\(x\)\(=\)\(5\ \text{or}\ -5\)
x equals 5 or negative 5
Example 2 — Shifted
Solve \(|x-3|=7\).
Solution

Set the inside to \(\pm7\).

\(x-3=7\)\(\Rightarrow\)\(x=10\)
\(x-3=-7\)\(\Rightarrow\)\(x=-4\)
x equals 10 or negative 4
Example 3 — Coefficient inside
Solve \(|2x+1|=9\).
Solution

Two cases, then solve each.

\(2x+1=9\)\(\Rightarrow\)\(x=4\)
\(2x+1=-9\)\(\Rightarrow\)\(x=-5\)
x equals 4 or negative 5
Example 4 — No solution
Solve \(|x|=-2\).
Solution

Absolute value is never negative.

\(|x|\)\(=\)\(-2\ \text{impossible}\)
No solution.
no solution, since absolute value cannot be negative

Common pitfalls

Write both cases — positive and negative.
Isolate the absolute value before splitting.
\(|X|=\)negative has no solution.

Frequently asked questions

How do you solve \(|x|=5\)?

\(x=5\) or \(x=-5\).

Why are there two solutions?

Two numbers, one positive and one negative, share the same distance from zero.

When does \(|X|=a\) have no solution?

When \(a\) is negative.

Do you isolate the absolute value first?

Yes, before splitting into cases.

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