Question

Solve each inequality. Check your solution.

$\mathbf{7}\mathit{w}\mathbf{\le }\mathbf{-}\mathbf{42}$

Short answer

The solution of the given inequality$7w\le -42$ is $\mathit{w}\mathbf{\le }\mathbf{-}\mathbf{6}$.

Step-by-step solution

Step 1. Write the division property of inequalities.

The division property of inequalities states that if both sides of the inequality are divided by a positive number the sign of the inequality remains the same and if both sides of the inequality are divided by a negative number then the sign of the inequality changes that is:

1. If $a>b$and$c$ is a positive number then $\frac{a}{c}>\frac{b}{c}$.
2. If $a<b$and$c$ is a positive number then $\frac{a}{c}<\frac{b}{c}$.
3. If$a>b$ and$c$ is a negative number then $\frac{a}{c}<\frac{b}{c}$.
4. If$a<b$ and$c$ is a negative number then $\frac{a}{c}>\frac{b}{c}$.

Step 2. Solve the given inequality 7w≤−42.

The solution of the given inequality$7w\le -42$ is:

$\begin{array}{c}7w\le -42\\ \frac{7w}{7}\le \frac{-42}{7}\left(\text{byusingdivisionpropertyofinequality}\right)\\ w\le -6\end{array}$

Therefore, the solution of the given inequality$7w\le -42$ is $w\le -6$.

Step 3. Check the solution.

To perform the check of the solution, substitute a number less than or equal to $-6$as $w\le -6$in the given inequality $7w\le -42$, if the condition obtained is true, the solution is correct and if the condition obtained is false, the solution is incorrect.

Let $w=-7$, as $-7\le -6$.

Now substitute $-7$for $w$in the inequality $7w\le -42$.

$\begin{array}{c}7w\le -42\\ 7\left(-7\right)\le -42\\ -49\le -42\end{array}$

As, the condition obtained$-49\le -42$ is true, therefore the solution is correct.