Question

Solve each inequality. Describe the solution set using set builder or interval notation. Then graph the solution set on a number line.

$\mathbf{-}\mathbf{12}<\mathbf{7}\mathbf{d}\mathbf{-}\mathbf{5}\le \mathbf{9}$

Short answer

Solution set of inequality is $\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{]}$.

Solution set on number line is

Step-by-step solution

Step 1. Write properties of inequality.

Properties of inequalities are written in following table:

Step 2. Solve -12<7d-5≤9.

To solve given inequality, use subtraction and division properties of inequality. Add 5 on each side then divide each side by 7 and simplify.

$\begin{array}{c}-12+5<7d-5+5\le 9+5\\ \frac{-7}{7}<\frac{7d}{7}\le \frac{14}{7}\\ -1<d\le 2\end{array}$

Thus, solution is $-1<d\le 2$.

Step 3. Describe solution set in interval notation.

Assolution set of the inequality is all real numbers greater than $-1$ or less than or equal to 2.

Write the solution set in interval notation.

$(-1,2]$

Step 4. Graph the solution set.

Open circle on $-1$ shows this point is excluded in the solution set and closed circle on 2 shows this point is included in the solution set.