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Algebra Absolute value equations and inequalities

Absolute value inequalities

19 practice questions 2 video lessons Theory + worked examples
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Theory

An absolute value inequality becomes a compound inequality:

  • Less than: \(|X|<a\ \Rightarrow\ -a<X<a\) (between, “and”).
  • Greater than: \(|X|>a\ \Rightarrow\ X<-a\) or \(X>a\) (outside, “or”).
Remember “less thAND, greatOR”: less-than is AND (between), greater-than is OR (outside).
Number line Absolute value less than 4 is the values between negative 4 and 4. -4 0 4 |x| < 4: between (-4, 4)
\(|x|<4\): the values between.
Number line Absolute value at least 3 is the values outside negative 3 and 3. -3 0 3 |x| ≥ 3: outside (x ≤ -3 or x ≥ 3)
\(|x|\ge3\): the values outside.

The two forms:

\[|X|<a\Rightarrow -a<X<a,\qquad |X|>a\Rightarrow X<-a\ \text{or}\ X>a\]
less than gives a between statement; greater than gives an outside statement
Isolate the absolute value first, then split.

How to solve

  1. Isolate the absolute value.
  2. Less-than \(\Rightarrow\) between; greater-than \(\Rightarrow\) outside.
  3. Write and solve the compound inequality.
  4. Graph on a number line.
Example 1 — Less than (between)
Solve \(|x|<4\).
Solution

“Less than” gives a between statement.

\(-4\)<\(x<4\)
negative 4 less than x less than 4
Example 2 — Greater than (outside)
Solve \(|x|\ge3\).
Solution

“Greater than” gives an outside (or) statement.

\(x\le-3\)\(\text{or}\)\(x\ge3\)
x at most negative 3 or at least 3
Example 3 — Shifted
Solve \(|x-2|<5\).
Solution

Between statement, then solve.

\(-5\)<\(x-2<5\)
\(-3\)<\(x<7\)
negative 3 less than x less than 7
Example 4 — Always true
Solve \(|x|>-1\).
Solution

Absolute value is always \(\ge0>-1\).

\(|x|\)\(>\)\(-1\ \text{always}\)
All real numbers.
all real numbers, since absolute value is always at least 0

Common pitfalls

Less-than is between (and); greater-than is outside (or).
Isolate the absolute value first.
\(|X|>\)negative is always true (all reals).

Frequently asked questions

What does \(|x|<4\) mean?

\(-4<x<4\) — the values between.

What does \(|x|>3\) mean?

\(x<-3\) or \(x>3\) — the values outside.

Which is 'and' and which is 'or'?

Less-than is “and” (between); greater-than is “or” (outside).

When is an absolute value inequality always true?

When it says \(|X|>\) a negative number.

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