How do you solve and graph abs(5-x)>=3?

Jul 27, 2017

See a solution process below:

Explanation:

The absolute value function takes any negative or positive 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.

$- 3 \ge 5 - x \ge 3$

First, subtract $\textcolor{red}{5}$ from each segment of the system of inequalities to isolate the $x$ term while keeping the system balanced:

$- \textcolor{red}{5} - 3 \ge - \textcolor{red}{5} + 5 - x \ge - \textcolor{red}{5} + 3$

$- 8 \ge 0 - x \ge - 2$

$- 8 \ge - x \ge - 2$

Now, multiply each segment by $\textcolor{b l u e}{- 1}$ to solve for $x$ while keeping the system balanced. However, because we are multiplying or dividing inequalities by a negative number we must reverse the inequality operators:

$\textcolor{b l u e}{- 1} \times - 8 \textcolor{red}{\le} \textcolor{b l u e}{- 1} \times - x \textcolor{red}{\le} \textcolor{b l u e}{- 1} \times - 2$

$8 \textcolor{red}{\le} x \textcolor{red}{\le} 2$

Or

$x \le 2$; $x \ge 8$

Or, in interval notation:

$\left(- \infty , 2\right]$; $\left[8 , + \infty\right)$