Question

Solve each inequality. Check your solution.

$\frac{t}{3}-6>-4$

Short answer

The solution of the given inequality$\frac{t}{3}-6>-4$ is $t>6$.

Check:

To perform the check of the solution, substitute a number greater than 6 as$t>6$in the given inequality $\frac{t}{3}-6>-4$, if the condition obtained is true, the solution is correct and if the condition obtained is false, the solution is incorrect.

Let $t=9$, as $9>6$.

Now substitute 9 for t in the inequality $\frac{t}{3}-6>-4$.

$$\begin{gathered} \begin{array}{c}\frac{t}{3}-6>-4\\ \frac{9}{3}-6>-4\\ 3-6>-4\end{array} \\ -3>-4 \end{gathered}$$

As, the condition obtained$-3>-4$ is true, therefore the solution is correct.

Step-by-step solution

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

The addition property of inequalities states that if the same number is added to each side of a true inequality, the resulting inequality is also true that is:

(i) If $a>b$, then $a+c>b+c$.

(ii) 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:

(i) If $a>b$andc is a positive number then $ac>bc$.

(ii) If $a<b$ and c is a positive number then $ac<bc$.

(ii) If $a>b$ and c is a negative number then $ac<bc$.

(iv) If $a<b$ and c is a negative number then $ac>bc$.

Step 2. Solve the given inequality t3−6>−4.

The solution of the given inequality$\frac{t}{3}-6>-4$is:

$$\begin{gathered} \frac{t}{3}-6>-4 \\ \frac{t}{3}-6+6>-4+6\left(\text{byusingadditionpropertyofinequality}\right) \\ \frac{t}{3}>2 \\ \left(3\right)\left(\frac{t}{3}\right)>\left(3\right)\left(2\right)\left(\text{byusingmutiplicationpropertyofinequality}\right) \\ t>6 \end{gathered}$$

Therefore, the solution of the given inequality$\frac{t}{3}-6>-4$is width="31">t>6.

Step 3. Check the solution.

To perform the check of the solution, substitute a number greater than 6 as$t>6$ in the given inequality $\frac{t}{3}-6>-4$, if the condition obtained is true, the solution is correct and if the condition obtained is false, the solution is incorrect.

Let $t=9$, as $9>6$.

Now substitute 9 for t in the inequality $\frac{t}{3}-6>-4$.

$$\begin{gathered} \begin{array}{c}\frac{t}{3}-6>-4\\ \frac{9}{3}-6>-4\\ 3-6>-4\end{array} \\ -3>-4 \end{gathered}$$

As, the condition obtained$-3>-4$ is true, therefore the solution is correct.