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
Texas Algebra II (TEKS) • Standard 2A.6(D), 2A.6(E), 2A.6(F) • Other Functions
Absolute Value Equations and Inequalities is the opening topic of Other Functions in the Texas Essential Knowledge and Skills (Algebra II, §111.40). It is aligned to Standard 2A.6(D), 2A.6(E), 2A.6(F), which requires students to formulate and solve absolute value linear equations and inequalities.
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)