# How do you solve the inequality -5q+9>24?

May 22, 2018

See a solution process below:

#### Explanation:

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

$- 5 q + 9 - \textcolor{red}{9} > 24 - \textcolor{red}{9}$

$- 5 q + 0 > 15$

$- 5 q > 15$

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

$\frac{- 5 q}{\textcolor{b l u e}{- 5}} \textcolor{red}{<} \frac{15}{\textcolor{b l u e}{- 5}}$

$\frac{\textcolor{b l u e}{\cancel{\textcolor{b l a c k}{- 5}}} q}{\cancel{\textcolor{b l u e}{- 5}}} \textcolor{red}{<} - 3$

$q \textcolor{red}{<} - 3$

May 22, 2018

$q < - 3$.

#### Explanation:

Solving an inequality is almost exactly like solving an equality, and for the most part you can treat it as such while solving it, except for one additional rule: whenever you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. For example, $>$ would go to $<$, $\le$ to $\ge$ and vice versa. If you wish to know why you must do this, read the next paragraph; otherwise, you may skip it.

The reason this rule arises is because of how the number line works. Observe that on the standard number line, numbers go smallest ($- \infty$) to largest ($\infty$) from left to right, with $0$ in the exact center. If we write $a < b$ we mean to say that $a$ is farther to the right than $a$. But, if we consider $- a$ and $- b$, we will notice that $- a < - b$ is false because $- a$ is farther to the right than $- b$.

$- 5 q + 9 > 24$.

First we subtract $9$ from both sides to get,

$- 5 q + 9 - 9 > 24 - 9 \Rightarrow - 5 q > 15$.

Now we divide both sides by $- 5$, flipping the inequality:

$\frac{- 5 q}{-} 5 > \frac{15}{-} 5 \Rightarrow q < - 3$.

May 22, 2018

$q < - 3$

#### Explanation:

$\text{isolate "-5q" by subtracting 9 from both sides}$

$\Rightarrow - 5 q > 24 - 9$

$\Rightarrow - 5 q > 15$

$\text{divide both sides by } - 5$

$\textcolor{b l u e}{\text{remember to reverse the sign as a consequence}}$

$\Rightarrow q < - 3$