Q44. - Algebra 2

Question

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

$\mathbf{2}\mathbf{-}\mathbf{3}\mathbf{z}\ge \mathbf{7}\left(\mathbf{8}\mathbf{-}\mathbf{2}\mathbf{z}\right)\mathbf{+}\mathbf{12}$

Step-by-Step Solution

Answer

Solution set of inequality is $\mathbf{[}\mathbf{6}\mathbf{,}\mathbf{+}\infty \mathbf{)}$.

Solution set on number line is 

Step 1 - Write properties of inequality.

Properties of inequalities are written in following table:

Step 2 - Solve 2 − 3 z ≥ 7 8 − 2 z + 12 .

To solve given inequality, use subtraction and division properties of inequality. Expand the right-hand side. Subtract 2 from each side then add $14z$ on each side and divide each side by 11 and simplify.

$\begin{array}{c}2-3z\ge 56-14z+12\\ 2-3z-2\ge 68-14z-2\\ -3z+14z\ge 66-14z+14z\\ \frac{11z}{11}\ge \frac{66}{11}\\ z\ge 6\end{array}$

Thus, solution is $z\ge 6$.

Step 3 – Describe solution set in interval notation.

As solution set of the inequality is all real numbers greater than or equal to 6.

Write the solution set in interval notation.

$[6,+\infty )$

Step 4 – Graph the solution set.

Closed circle on 6 shows this point is included in the solution set.