# 4 Important Types of Absolute Value Equations

There are 4 main types of absolute value equations regarding whether there are;

1. absolute value and a static value
2. absolute value and an expression involving unknown pronumerals
3. two absolute values in both sides
4. two absolute values and a value

## Type 1: One Absolute Value and a Constant

Solve $| x-2 | = 5$.

\begin{aligned} \displaystyle x-2 = 5 &\text{ or } x-2 = -5 \\ \therefore x = 7 &\text{ or } x = -3 \end{aligned}

## Type 2: One Absolute Value and a Linear Expression

Solve $| x-1 | = 2x+4$.

\begin{aligned} \displaystyle \require{AMSsymbols} \require{color} (|x-1|)^2 &= (2x+4)^2 &\color{red} \text{squaring both sides} \\ (x-1)^2 &= (2x+4)^2 \\ x^2-2x + 1 &= 4x^2 + 16x + 16 \\ 3x^2 + 18x + 15 &= 0 \\ x^2 + 6x + 5 &= 0 \\ (x+5)(x+1) &= 0 \\ x = -5 &\text{ or } x = -1 \\ \color{red} \text{Test } x = -5 \\ \text{LHS} &= |-5-1| \\ &= |-6| \\ &= 6 \\ \text{RHS} &= 2 \times -5 +4 \\ &= -6 \\ \text{LHS} &\ne \text{RHS} \\ \therefore x &\ne -5 \\ \color{red} \text{Test } x = -1 \\ \text{LHS} &= |-1-1| \\ &= |-2| \\ &= 2 \\ \text{RHS} &= 2 \times -1 +4 \\ &= 2 \\ \text{LHS} &= \text{RHS} \\ \therefore x &= -1 \end{aligned}

## Type 3: Two Absolute Value Expressions

Solve $| x-1 | = |3-x|$.

\begin{aligned} \displaystyle \require{AMSsymbols} \require{color} x-1 &= \pm(3-x) \\ x-1 &= 3-x \color{red} \cdots (1) \\ 2x &= 4 \\ x &= 2 \\ x-1 &= -3+x \color{red} \cdots (2) \\ x-x &= -2 \\ 0 &= -2 &\color{red} \text{no solution} \\ \therefore x &= 2 &\color{red} \text{from (a) and (2)} \end{aligned}

## Type 4: Two Absolute Value Expressions and a Constant

Solve $| x+2 | + |x-3| = 7$.

\begin{aligned} \displaystyle \require{AMSsymbols}\require{color} -(x + 2)-(x-3) &= 7 &\color{red} \text{for } x \lt -2 \\ -x-2-x+3 &= 7 \\ -2x &= 6 \\ x &= -3 &\color{red} \text{this is OK for } x \lt -2 \\ (x + 2)-(x-3) &= 7 &\color{red} \text{for } -2 \le x \lt 3 \\ x+2-x+3 &= 7 \\ 5 &\ne 7 &\color{red} \text{ no solution}\\ (x + 2) + (x-3) &= 7 &\color{red} \text{for } x \ge 3 \\ x+2+x-3 &= 7 \\ 2x &= 8 \\ x &= 4 &\color{red} \text{this is OK for } x \ge 3 \\ \therefore x &= -3 \text{ or } 4 \end{aligned}