Question

Solve each inequality. Check your solution.

$\mathbf{-}\frac{\mathbf{2}}{\mathbf{3}}\mathit{t}\mathbf{\ge }\mathbf{24}$

Short answer

The solution of the given inequality$-\frac{2}{3}t\ge 24$is $\mathit{t}\mathbf{\le }\mathbf{-}\mathbf{36}$.

Step-by-step solution

Step 1. Write the multiplication property of inequalities.

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 −23t≥24.

The solution of the given inequality$-\frac{2}{3}t\ge 24$ is:

$\begin{array}{c}-\frac{2}{3}t\ge 24\\ \left(\frac{-3}{2}\right)\left(-\frac{2}{3}t\right)\le \left(\frac{-3}{2}\right)\left(24\right)\left(\text{byusingmutiplicationpropertyofinequality}\right)\\ \left(\frac{-3}{2}\times \frac{-2}{3}\right)t\le -\frac{3}{2}\times 24\\ t\le -36\end{array}$

Therefore, the solution of the given inequality$-\frac{2}{3}t\ge 24$ is $t\le -36$.

Step 3. Check the solution.

To perform the check of the solution, substitute a number less than or equal to $-36$as $t\le -36$in the given inequality $-\frac{2}{3}t\ge 24$, if the condition obtained is true, the solution is correct and if the condition obtained is false, the solution is incorrect.

Let $t=-39$, as $-39\le -36$.

Now substitute $-39$for $t$in the inequality $-\frac{2}{3}t\ge 24$.

$\begin{array}{c}-\frac{2}{3}t\ge 24\\ \frac{-2}{3}\left(-39\right)\ge 24\\ -2\times -13\ge 24\\ 26\ge 24\end{array}$

As, the condition obtained$26\ge 24$ is true, therefore the solution is correct.