How do you graph and solve  | 4 - 2x | - 9 <=-3?

1 Answer
Jun 13, 2017

See a solution process below:

Explanation:

First. add $\textcolor{red}{9}$ to each side of the inequality to isolate the absolute value function while keeping the inequality balanced:

$\left\mid 4 - 2 x \right\mid - 9 + \textcolor{red}{9} \le - 3 + \textcolor{red}{9}$

$\left\mid 4 - 2 x \right\mid - 0 \le 6$

$\left\mid 4 - 2 x \right\mid \le 6$

The absolute value function takes any negative or positive term and transforms it to its positive form. Therefore, we must solve the term within the absolute value function for both its negative and positive equivalent.

$- 6 \le 4 - 2 x \le 6$

Next, subtract $\textcolor{red}{4}$ from each segment of the system of inequalities to isolate the $x$ term while keeping the system balanced:

$- \textcolor{red}{4} - 6 \le - \textcolor{red}{4} + 4 - 2 x \le - \textcolor{red}{4} + 6$

$- 10 \le 0 - 2 x \le 2$

$- 10 \le - 2 x \le 2$

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

$\frac{- 10}{\textcolor{b l u e}{- 2}} \textcolor{red}{\ge} \frac{- 2 x}{\textcolor{b l u e}{- 2}} \textcolor{red}{\ge} \frac{2}{\textcolor{b l u e}{- 2}}$

$5 \textcolor{red}{\ge} \frac{\textcolor{b l u e}{\cancel{\textcolor{b l a c k}{- 2}}} x}{\cancel{\textcolor{b l u e}{- 2}}} \textcolor{red}{\ge} - 1$

$5 \textcolor{red}{\ge} x \textcolor{red}{\ge} - 1$

Or

$x \ge - 1$ and $x \le 5$

Or, in interval notation:

$\left[- 1 , 5\right]$