Other Types of Equations: Learn It 4

Absolute Value Equations

An absolute value equation is an equation in which the variable of interest is contained within absolute value bars. Absolute value is defined as a distance. That is, the bars are used to designate that the number inside the absolute value represents its distance from zero on the number line.

absolute value

The absolute value of a number is a measure of its distance from zero on the number line, regardless of direction. It is always a non-negative value.

 

For a real number [latex]x[/latex], the absolute value is denoted by [latex]|x|[/latex] and is defined as:

[latex]|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}[/latex]

The absolute value of both [latex]-3[/latex] and [latex]3[/latex] is [latex]3[/latex] because both are three units away from zero on the number line.
In absolute value notation:

[latex]|-3| = 3[/latex] and [latex]|3| = 3[/latex]

An absolute value equation is an equation that contains an absolute value expression.

absolute value equation

The general form of an absolute value equation is:

[latex]|A| = B[/latex]

where [latex]A[/latex] is the expression and [latex]B[/latex] is a non-negative number.

 

  • For real numbers [latex]A[/latex] and [latex]B[/latex], an equation of the form [latex]|A|=B[/latex], with [latex]B\ge 0[/latex], will have solutions when [latex]A=B[/latex] or [latex]A=-B[/latex]. If [latex]B<0[/latex], the equation [latex]|A|=B[/latex] has no solution.

 

  • An absolute value equation in the form [latex]|ax+b|=c[/latex] has the following properties:
[latex]\begin{array}{l}\text{If }c<0,|ax+b|=c\text{ has no solution}.\hfill \\ \text{If }c=0,|ax+b|=c\text{ has one solution}.\hfill \\ \text{If }c>0,|ax+b|=c\text{ has two solutions}.\hfill \end{array}[/latex]
How To: Given an absolute value equation, solve it

  1. Isolate the Absolute Value Expression:
    Start by getting the absolute value expression by itself on one side of the equation. This typically involves simplifying the equation and moving any terms that are not inside the absolute value to the other side.
  2. Write Two Separate Equations:
    Since the absolute value of a number [latex]x[/latex] an either be [latex]x[/latex] itself if [latex]x[/latex] is positive or [latex]-x[/latex]if [latex]x[/latex] is negative, split the equation into two cases:

    • Case 1: The expression inside the absolute value equals the value on the other side.
    • Case 2: The expression inside the absolute value equals the negative of the value on the other side.
  3. Solve Each Equation Separately:
    Solve the two equations that result from step 2. This may involve further simplifications or solving quadratic equations, depending on the form of the original equation.
  4. Check for Extraneous Solutions:
    Since absolute value equations can introduce solutions that do not actually satisfy the original equation when plugged back in, it’s crucial to substitute each found solution back into the original equation to verify its validity.
Solve the absolute value equation:

[latex]|6x+4|=8[/latex]
[latex]\begin{align*} \text{Original equation} & : & |6x + 4| &= 8 \\ \text{Remove the absolute value, consider both cases} & : & 6x + 4 &= 8 & 6x + 4 &= -8 \\ \text{Subtract 4 from both sides} & : & 6x &= 4 & 6x &= -12 \\ \text{Divide both sides by 6} & : & x &= \frac{4}{6} & x &= \frac{-12}{6} \\ \text{Simplify each fraction} & : & x &= \frac{2}{3} & x &= -2 \\ \end{align*}[/latex]

The two solutions are [latex]x=\dfrac{2}{3}[/latex], [latex]x=-2[/latex].

Solve the absolute value equations:

  1. [latex]|3x+4|=-9[/latex]

  2. [latex]|3x - 5|-4=6[/latex]

  3. [latex]|-5x+10|=0[/latex]

When solving absolute value equations, the absolute value expression must be isolated on one side of the equation before setting up the two cases to remove the absolute value bars.
[latex]\\[/latex]
Use the properties of equality to isolate the absolute value expression, but avoid multiplying into or dividing from any expression inside the bars.