Question

Solve each inequality. Check your solution.

$\frac{\mathbf{g}}{\mathbf{4}}\mathbf{+}\mathbf{3}\mathbf{\le }\mathbf{-}\mathbf{9}$

Short answer

The solution of the given inequality$\frac{g}{4}+3\le -9$ is $\mathit{g}\mathbf{\le }\mathbf{-}\mathbf{48}$.

Step-by-step solution

Step 1. Write the subtraction and multiplication property of inequalities.

The subtraction property of inequalities states that if the same number is subtracted from each side of a true inequality, the resulting inequality is also true that is:

1. If $a>b$, then $a-c>b-c$.
2. If $a<b$, then $a-c<b-c$.

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

1. If$a>b$ and$c$ is a positive number then $ac>bc$.
2. If$a<b$ and$c$ is a positive number then $ac<bc$.
3. If$a>b$ and$c$ is a negative number then $ac<bc$.
4. If$a<b$ and$c$ is a negative number then $ac>bc$.

Step 2. Solve the given inequality g4+3≤−9.

The solution of the given inequality $\frac{g}{4}+3\le -9$is:

$\begin{array}{c}\frac{g}{4}+3\le -9\\ \frac{g}{4}+3-3\le -9-3\left(\text{byusingsubtractionpropertyofinequality}\right)\\ \frac{g}{4}\le -12\\ \left(4\right)\left(\frac{g}{4}\right)\le \left(4\right)\left(-12\right)\left(\text{byusingmultiplicationpropertyofinequality}\right)\\ g\le -48\end{array}$

Therefore, the solution of the given inequality$\frac{g}{4}+3\le -9$ is $g\le -48$.

Step 3. Check the solution.

To perform the check of the solution, substitute a number less than or equal to $-48$as $g\le -48$in the given inequality $\frac{g}{4}+3\le -9$, if the condition obtained is true, the solution is correct and if the condition obtained is false, the solution is incorrect.

Let $g=-52$, as $-52\le -48$.

Now substitute $-52$for $g$in the inequality $\frac{g}{4}+3\le -9$.

$\begin{array}{c}\frac{g}{4}+3\le -9\\ \frac{-52}{4}+3\le -9\\ -13+3\le -9\\ -10\le -9\end{array}$

As, the condition obtained$-10\le -9$ is true, therefore the solution is correct.