Question

Solve each inequality. Check your solution.$\frac{\mathbf{1}}{\mathbf{3}}\mathit{x}\mathbf{>}\mathbf{6}$

Short answer

The solution of the given inequality $\frac{1}{3}x>6$is role="math" localid="1647428447602" $x>18$.

Check:

To perform the check of the solution, substitute a number greater than 18 as $x>18$in the given inequality $\frac{1}{3}x>6$, if the condition obtained is true, the solution is correct and if the condition obtained is false, the solution is incorrect.

Let, $x=21$as $21>18$.

Now substitute 21 for $x$in the inequality $\frac{1}{3}x>6$

$\begin{array}{c}\frac{1}{3}x>6\\ \frac{1}{3}\left(21\right)>6\\ 7>6\end{array}$

As, the condition obtained $7>6$is true, therefore the solution is correct.

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:

(i) If$a>b$and$c$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 .

The solution of the given inequality $\frac{1}{3}x>6$is:

$\begin{array}{c}\frac{1}{3}x>6\\ \left(3\right)\left(\frac{1}{3}x\right)>\left(3\right)\left(\text{6}\right)\left(\text{byusingmutiplicationpropertyofinequality}\right)\\ x>18\end{array}$

Therefore, the solution of the given inequality $\frac{1}{3}x>6$is$x>18$.

Step 3. Check the solution.

To perform the check of the solution, substitute a number greater than 18 as $x>18$in the given inequality $\frac{1}{3}x>6$, if the condition obtained is true, the solution is correct and if the condition obtained is false, the solution is incorrect.

Let $x=21$, as $21>18$.

Now substitute 21 for $x$in the inequality $\frac{1}{3}x>6$.

$\begin{array}{c}\frac{1}{3}x>6\\ \frac{1}{3}\left(21\right)>6\\ 7>6\end{array}$

As, the condition obtained$7>6$ is true, therefore the solution is correct.