Exercise 41 Page 33

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 $\text{the}$ $\text{opposites}$ $\text{of}$ $\text{the}$ $\text{expressions}$ $\text{are}$ $\text{equal}.$ Now let's consider these two cases for the given equation. Let's solve for $h$ in each of these equations simultaneously. After solving an absolute value equation, it is necessary to check for extraneous solutions. To do this, we substitute the found solutions into the given equation and determine if a true statement is made. We end with a true statement, so $h=\frac{1}{4}$ is not extraneous. We can write $h=\frac{1}{4}$ as $h=0.25.$ We end with a false statement, so $h=\text{-}\frac{1}{10}$ is extraneous. Therefore, it is not a valid solution to the equation.