An absolute value equation is an equation containing an expression inside absolute value bars.
The absolute value of a number is its distance from zero, so it is never negative.
To solve absolute value equations, we usually split the problem into cases.
1. Absolute value definition
For any real expression \(A\), absolute value is defined by:
\[
\begin{aligned}
|A|=
\begin{cases}
A, & A\ge 0,\\
-A, & A<0.
\end{cases}
\end{aligned}
\]
This definition is the reason absolute value equations often produce two possible equations.
2. Simple absolute value equation
If:
\[
\begin{aligned}
|A|=c
\end{aligned}
\]
and \(c>0\), then:
\[
\begin{aligned}
A=c
\qquad\text{or}\qquad
A=-c.
\end{aligned}
\]
If \(c=0\), then \(A=0\). If \(c<0\), there is no real solution because an absolute value cannot be negative.
3. Worked example
Solve:
\[
\begin{aligned}
|2x-5|=7.
\end{aligned}
\]
Split into two cases:
\[
\begin{aligned}
2x-5&=7
\qquad\text{or}\qquad
2x-5=-7.
\end{aligned}
\]
Solve the first case:
\[
\begin{aligned}
2x-5&=7,\\
2x&=12,\\
x&=6.
\end{aligned}
\]
Solve the second case:
\[
\begin{aligned}
2x-5&=-7,\\
2x&=-2,\\
x&=-1.
\end{aligned}
\]
Verify:
\[
\begin{aligned}
|2(6)-5|&=|7|=7,\\
|2(-1)-5|&=|-7|=7.
\end{aligned}
\]
Therefore:
\[
\begin{aligned}
\boxed{x=6\text{ or }x=-1}.
\end{aligned}
\]
4. Why case splitting works
The expression inside the absolute value can be positive or negative.
For \(|2x-5|\), the sign changes when:
\[
\begin{aligned}
2x-5&=0,\\
x&=\frac{5}{2}.
\end{aligned}
\]
For \(x\ge 5/2\), \(|2x-5|=2x-5\).
For \(x<5/2\), \(|2x-5|=-(2x-5)\).
5. Multiple absolute values
Equations such as:
\[
\begin{aligned}
|x-1|+|x+1|=4
\end{aligned}
\]
have more than one breakpoint. The breakpoints occur where each absolute-value input is zero:
\[
\begin{aligned}
x-1&=0 \quad\Rightarrow\quad x=1,\\
x+1&=0 \quad\Rightarrow\quad x=-1.
\end{aligned}
\]
These breakpoints divide the number line into intervals:
\[
\begin{aligned}
(-\infty,-1),\qquad (-1,1),\qquad (1,\infty).
\end{aligned}
\]
On each interval, every absolute-value expression has a fixed sign, so the equation becomes an ordinary linear equation.
6. Equations with intervals of solutions
Some absolute value equations have infinitely many solutions. For example:
\[
\begin{aligned}
|x|=x.
\end{aligned}
\]
Since \(|x|=x\) for every \(x\ge0\), the solution is:
\[
\begin{aligned}
\boxed{x\in[0,\infty)}.
\end{aligned}
\]
A number line is the clearest way to display this kind of solution.
7. Graph interpretation
An absolute value equation:
\[
\begin{aligned}
\text{LHS}(x)=\text{RHS}(x)
\end{aligned}
\]
can be graphed as two curves:
\[
\begin{aligned}
y_1&=\text{LHS}(x),\\
y_2&=\text{RHS}(x).
\end{aligned}
\]
Point solutions occur where the curves intersect. Interval solutions occur where the two graphs overlap along a segment.
8. Verification
Every candidate should be checked in the original equation:
\[
\begin{aligned}
\text{LHS}(x_{\text{candidate}})
\stackrel{?}{=}
\text{RHS}(x_{\text{candidate}}).
\end{aligned}
\]
Verification catches sign-case mistakes and numerical false positives.
9. Formula summary
The table uses plain-text formulas in the cells to avoid raw LaTeX issues in narrow layouts.
10. Common mistakes
- Solving only \(A=c\) and forgetting the \(A=-c\) case.
- Claiming a solution for \(|A|=-c\) when \(c>0\).
- Forgetting to check candidates in the original equation.
- Using the wrong sign on a case interval.
- Missing interval solutions such as \(|x|=x\).
- Reading a graph intersection without verifying the algebra.
Key idea: absolute value equations are solved by splitting signs, solving each case, and verifying the final solution set.
