Equations with absolute value

To solve an equation with an absolute value you remove the modulus by studying the sign of its argument, and you check that every solution found is compatible with the region in which it was obtained.

Property — How to solve $|f(x)|=k$

To solve an equation with an absolute value $|f(x)|=k$:

• study the sign of the argument $f(x)$;
• solve the equation in each region, removing the absolute value with the appropriate sign;
• accept only the solutions compatible with the region considered.

Example

$|4-x^2|-12=0$, that is $|4-x^2|=12$. Two cases:

1. $4-x^2=12 \implies x^2=-8$: impossible;
2. $4-x^2=-12 \implies x^2=16 \implies \boxed{x=\pm4}$.

Example

$|x^2-5x+6|=2x$. The argument $x^2-5x+6=(x-2)(x-3)$ has zeros at $x=2$ and $x=3$.

Case 1 ($x\le2$ or $x\ge3$, where the argument is $\ge0$): $x^2-5x+6=2x \implies x^2-7x+6=0 \implies (x-6)(x-1)=0,$ whence $x=6$ (in the region $x\ge3$) and $x=1$ (in the region $x\le2$): both acceptable.

Case 2 ($2<x<3$, where the argument is $<0$): $-(x^2-5x+6)=2x \implies -x^2+3x-6=0 \implies \Delta=9-24<0,$ impossible.

Solutions: $\boxed{x=1}$ and $\boxed{x=6}$.

Example — Equation with two absolute values in the plane

$|y-x|+|y+2|-4=0$. The lines $y=x$ and $y=-2$ separate the plane into four regions:

In each region the absolute value is removed with the appropriate sign and the resulting linear equation is solved. Each region produces a segment; the four pieces form a rhombus.

The lines $y=x$ and $y=-2$ divide the plane into four regions; in each of them the moduli are removed with different signs.

Links

Topics: Parabola
Concepts: Absolute value
Skills: Reasoning by cases · Solving equations