How do you solve and graph abs(-k-7)>=4?

Nov 22, 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.

$- 4 \ge - k - 7 \ge 4$

First, add $\textcolor{red}{7}$ to each segment of the system of inequalities to isolate the $k$ term while keeping the system balanced:

$- 4 + \textcolor{red}{7} \ge - k - 7 + \textcolor{red}{7} \ge 4 + \textcolor{red}{7}$

$3 \ge - k - 0 \ge 11$

$3 \ge - k \ge 11$

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

$\textcolor{b l u e}{- 1} \times 3 \textcolor{red}{\le} \textcolor{b l u e}{- 1} \times - k \textcolor{red}{\le} \textcolor{b l u e}{- 1} \times 11$

$- 3 \textcolor{red}{\le} k \textcolor{red}{\le} - 11$

Or

$k \le - 11$; $k \ge - 3$

Or, in interval notation:

$\left(- \infty , - 11\right]$; $\left[- 3 , + \infty\right)$