# How do you solve and graph abs(3-2r)>7?

Dec 11, 2017

See a solution process below:

#### Explanation:

The absolute value function takes any term and transforms it to its non-negative form. Therefore, we must solve the term within the absolute value function for both its negative and positive equivalent. Therefore we can rewrite this problem as:

$- 7 > 3 - 2 r > 7$

First, subtract $\textcolor{red}{3}$ from each segment of the system of inequalities to isolate the $r$ term while keeping the system balanced:

$- 7 - \textcolor{red}{3} > 3 - \textcolor{red}{3} - 2 r > 7 - \textcolor{red}{3}$

$- 10 > 0 - 2 r > 4$

$- 10 > - 2 r > 4$

Now divide each segment by $\textcolor{b l u e}{- 2}$ to solve for $r$ while keeping the system balanced. However, because we are multiplying or dividing inequalities by a negative number we must reverse the inequality operators:

$- \frac{10}{\textcolor{b l u e}{- 2}} \textcolor{red}{<} \frac{- 2 r}{\textcolor{b l u e}{- 2}} \textcolor{red}{<} \frac{4}{\textcolor{b l u e}{- 2}}$

$5 \textcolor{red}{<} \frac{\textcolor{b l u e}{\cancel{\textcolor{b l a c k}{- 2}}} r}{\cancel{\textcolor{b l u e}{- 2}}} \textcolor{red}{<} - 2$

$5 \textcolor{red}{<} r \textcolor{red}{<} - 2$

Or

$r < - 2$; $r > 5$

Or, in interval notation:

(-oo, -2); (5, +oo)