Algebra 2
Other functions (advanced)
Absolute value equations and inequalities (advanced)
20 practice questions
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Theory + worked examples
Absolute Value Equations and Inequalities
California Algebra 2 • Standard A-REI.11 • Other Functions
Absolute Value Equations and Inequalities is the opening topic of Other Functions in the California Common Core State Standards. It is aligned to Standard A-REI.11, which requires students to solve absolute value equations and inequalities and represent their solutions.
Absolute value equations split into two cases; inequalities become “between” for less-than and “outside” for greater-than.
Theory
Absolute value measures distance from zero, so it produces two cases:
- Equation: \(|X|=a\Rightarrow X=a\) or \(X=-a\).
- Less than: \(|X|<a\Rightarrow -a<X<a\) (between).
- Greater than: \(|X|>a\Rightarrow X<-a\) or \(X>a\) (outside).
Isolate the absolute value first, and remember \(|X|=a\) has no solution if \(a<0\).
\(y=|x-1|\) has its vertex where the inside is zero.
The three absolute-value cases.
The case split:
\[|X|=a\Rightarrow X=\pm a,\quad |X|<a\Rightarrow -a<X<a\]
Less than = and (between); greater than = or (outside).
How to solve
- Isolate the absolute value.
- For \(=\), write two equations with \(\pm\).
- For \(<\), write a between statement; for \(>\), an outside statement.
- Solve each part.
Example 1 — Basic equation
Solve \(|x|=5\).
Solution
The inside is \(5\) or \(-5\).
| \(x\) | \(=\) | \(5\ \text{or}\ -5\) |
Example 2 — Shifted equation
Solve \(|x-3|=7\).
Solution
Set the inside to \(\pm7\).
| \(x-3=7\) | \(\Rightarrow\) | \(x=10\) |
| \(x-3=-7\) | \(\Rightarrow\) | \(x=-4\) |
Example 3 — Less-than inequality
Solve \(|x|<4\).
Solution
“Less than” gives a between statement.
| \(-4\) | < | \(x<4\) |
Example 4 — Greater-than inequality
Solve \(|x-2|\ge 3\).
Solution
“Greater than” gives an outside (or) statement.
| \(x-2\ge3\) | \(\Rightarrow\) | \(x\ge5\) |
| \(x-2\le-3\) | \(\Rightarrow\) | \(x\le-1\) |
Common pitfalls
Isolate the absolute value first.
Less-than is between; greater-than is outside — don't swap.
\(|X|=\)negative has no solution.
Frequently asked questions
How do you solve \(|x|=5\)?
\(x=5\) or \(x=-5\).
What does \(|x|<4\) mean?
\(-4<x<4\) — the values between.
What does \(|x|>4\) mean?
\(x<-4\) or \(x>4\) — the values outside.
When does an absolute value equation have no solution?
When it equals a negative number.
More in Other functions (advanced)