Exercise 51 Page 33

Consider an absolute value equation of the following form. The equation can have either zero, one or two solutions. How many, depends on the value of $c.$

• $\mathbf{c}<0\text{:}$ Since an absolute value of something cannot be negative, there exists no $x$ that would satisfy the equation. Therefore, if $c$ is negative, there are no solutions.
• $\mathbf{c}=0\text{:}$ If an absolute value of an expression equals $0,$ the expression itself equals $0.$ This means we will have $1$ solution.
• $\mathbf{c}>0\text{:}$ If $c$ is a positive value, we will have two solutions. This is because when you remove the absolute value, $ax+b$ could both equal $c$ and $\text{-}c.$

Classifying the Given Equations

We will use above information to classify the equations given in the exercise. By isolating the absolute value expression in each equation we can determine the number of solutions by looking at the value of $c.$

Equation	$|ax+b|=c$	$c\text{ is}\dots$	Solutions
$|x-2|+6=0$	$|x-2|=\text{-}6$	Negative	$0$
$|x+8|+2=7$	$|x+8|=5$	Positive	$2$
$|x-6|-5=\text{-}9$	$|x-6|=\text{-}4$	Negative	$0$
$|x+3|-1=0$	$|x+3|=1$	Positive	$2$
$|x-1|+4=4$	$|x-1|=0$	Zero	$1$
$|x+5|-8=\text{-}8$	$|x+5|=0$	Zero	$1$