# How do you graph and solve  | 3 - 7 x |>17?

Jan 23, 2018

See a solution process below: (-2; 20/7)

#### 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.

We can rewrite and solve this inequality as:

$- 17 > 3 - 7 x > 17$

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

$- 17 - \textcolor{red}{3} > 3 - \textcolor{red}{3} - 7 x > 17 - \textcolor{red}{3}$

$- 20 > 0 - 7 x > 14$

$- 20 > - 7 x > 14$

Now, divide each segment by $\textcolor{b l u e}{- 7}$ to solve for $x$ while keeping the system balanced. Because we are multiplying or dividing an inequality by a negative number we must reverse the inequality operators.

$\frac{- 20}{\textcolor{b l u e}{- 7}} \textcolor{red}{<} \frac{- 7 x}{\textcolor{b l u e}{- 7}} \textcolor{red}{<} \frac{14}{\textcolor{b l u e}{- 7}}$

$\frac{20}{7} \textcolor{red}{<} \frac{\textcolor{b l u e}{\cancel{\textcolor{b l a c k}{- 7}}} x}{\cancel{\textcolor{b l u e}{- 7}}} \textcolor{red}{<} - 2$

$\frac{20}{7} \textcolor{red}{<} x \textcolor{red}{<} - 2$

Or.

$x < - 2$; $x > \frac{20}{7}$

Or, in interval notation:

$\left(- \infty , - 2\right)$; $\left(\frac{20}{7} , + \infty\right)$

To graph this we will draw a vertical lines at $- 2$ and $\frac{20}{7}$ on the horizontal axis.

The lines will be a dashed lines because the inequality operators do not contain an "or equal to" clause.

We will shade outside the two lines to show the interval where the inequality is true.