Question

Solve each inequality. Check your solution.

$\mathbf{3}\mathit{c}\mathbf{-}\mathbf{7}\mathbf{<}\mathbf{11}$

Short answer

The solution of the given inequality$3c-7<11$ is $\mathit{c}\mathbf{<}\mathbf{6}$.

Step-by-step solution

Step 1. Write the addition and division 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:

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

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 3c−7<11.

The solution of the given inequality $3c-7<11$is:

$\begin{array}{c}3c-7<11\\ 3c-7+7<11+7\left(\text{byusingadditionpropertyofinequality}\right)\\ 3c<18\\ \frac{3c}{3}\text{<}\frac{18}{3}\left(\text{byusingdivisionpropertyofinequality}\right)\\ c<6\end{array}$

Therefore, the solution of the given inequality $3c-7<11$is $c<6$.

Step 3. Check the solution.

To perform the check of the solution, substitute a number less than 6 as$c<6$ in the given inequality $3c-7<11$, if the condition obtained is true, the solution is correct and if the condition obtained is false, the solution is incorrect.

Let $c=5$, as $5<6$.

Now substitute 5 for $c$in the inequality $3c-7<11$.

$\begin{array}{c}3c-7<11\\ 3\left(5\right)-7<11\\ 15-7<11\\ 8<11\end{array}$

As, the condition obtained$8<11$ is true, therefore the solution is correct.