# How do you solve the inequality -w - 4w + 9 ≤ w - 21 - w?

May 30, 2017

See a solution process below:

#### Explanation:

First, group and combine like terms on each side of the inequality:

$- w - 4 w + 9 \le w - 21 - w$

$- 1 w - 4 w + 9 \le w - w - 21$

$\left(- 1 - 4\right) w + 9 \le 0 - 21$

$- 5 w + 9 \le - 21$

Next, subtract $\textcolor{red}{9}$ from each side of the inequality to isolate the $w$ term while keeping the inequality balanced:

$- 5 w + 9 - \textcolor{red}{9} \le - 21 - \textcolor{red}{9}$

$- 5 w + 0 \le - 30$

$- 5 w \le - 30$

Now, divide each side of the inequality by $\textcolor{b l u e}{- 5}$ to solve for $w$ while keeping the inequality balanced. However, because we are multiplying or dividing an inequality by a negative number we need to reverse the inequality operator:

$\frac{- 5 w}{\textcolor{b l u e}{- 5}} \textcolor{red}{\ge} \frac{- 30}{\textcolor{b l u e}{- 5}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{- 5}}} w}{\cancel{\textcolor{b l u e}{- 5}}} \textcolor{red}{\ge} 6$

$w \ge 6$

May 30, 2017

$\omega \ge 6$

#### Explanation:

$- \omega - 4 \omega + 9 \le \omega - 21 - \omega \rightarrow - 5 \omega + 9 \le \cancel{\omega} - 21 - \cancel{\omega} \rightarrow - 5 \omega \le - 30 \rightarrow 5 \omega \ge 30 \rightarrow \omega \ge \frac{30}{5} = 6$