Exercise 29 Page 247

When solving an equation involving absolute value expressions, we should consider what would happen if we removed the absolute value symbols. Let's look at an example equation. Although we can make $4$ statements about this equation, there are actually only two possible cases to consider.

Statement	Result
Both absolute values are positive.	$ax+b=cx+d$
Both absolute values are negative.	$\text{-}(ax+b)=\text{-}(cx+d)$
Only the left-hand side is negative.	$\text{-}(ax+b)=cx+d$
Only the right-hand side is negative.	$ax+b=\text{-}(cx+d)$

Because of the Properties of Equality, when the absolute values of two expressions are equal, either $\text{the}$ $\text{expressions}$ $\text{are}$ $\text{equal}$ or $\text{the}$ $\text{opposites}$ $\text{of}$ $\text{the}$ $\text{expressions}$ $\text{are}$ $\text{equal}.$ First, notice that, using the properties of absolute value, we can rewrite the equation as shown below.