There are 4 main types of absolute value equations regarding whether there are;
- absolute value and a static value
- absolute value and an expression involving unknown pronumerals
- two absolute values in both sides
- 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 \( | x+2 | + |x-3| = 7 \).
\( \begin{aligned} \displaystyle \require{AMSsymbols}\require{color}
-(x + 2)-(x-3) &= 7 &\color{red} \text{for } x \lt -2 \\
-x-2-x+3 &= 7 \\
-2x &= 6 \\
x &= -3 &\color{red} \text{this is OK for } x \lt -2 \\
(x + 2)-(x-3) &= 7 &\color{red} \text{for } -2 \le x \lt 3 \\
x+2-x+3 &= 7 \\
5 &\ne 7 &\color{red} \text{ no solution}\\
(x + 2) + (x-3) &= 7 &\color{red} \text{for } x \ge 3 \\
x+2+x-3 &= 7 \\
2x &= 8 \\
x &= 4 &\color{red} \text{this is OK for } x \ge 3 \\
\therefore x &= -3 \text{ or } 4
\end{aligned} \)
Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorials Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Mathematical Induction Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume