Chapter 1.5, Problem 45PPSE

Solution

To solve the inequality $2-3z\ge 7\left(8-2z\right)+12$ and graph the solution.

  $z\ge 6$

The graph of solution set is shown below:

  

Given:

The compound inequality, $2-3z\ge 7\left(8-2z\right)+12$ .

Concept Used:

• If there is the strict inequality symbol $<\text{ or}>$ , then use a parenthesis $\left(\right)$ on the interval
• If there is the inequality symbol $\le \text{ or}\ge$ , then use a bracket $\left[\right]$ on the interval
• To graph the endpoint with strict inequality $<\text{ or}>$ , use the open parenthesis or a hollow circle at that endpoint.
• To graph the endpoint with inequality symbol $\le \text{ or}\ge$ , use the bracket or a solid circle at that endpoint.
• Multiplying or dividing an inequality by a negative number reverse the inequality sign.

Calculation:

In order to solve the given inequality $2-3z\ge 7\left(8-2z\right)+12$ , perform the basic algebraic operations to isolate the variable term as shown below

  $\begin{array}{c}2-3z\ge 7\left(8-2z\right)+12\\ 2-3z\ge 56-14z+12\\ 2-3z\ge 68-14z\\ 2-3z+14z\ge 68-14z+14z\\ 2+11z\ge 68\\ 2-2+11z\ge 68-2\\ 11z\ge 66\\ \frac{11z}{11}\ge \frac{66}{11}\\ z\ge 6\end{array}$

Thus, the solution of the given inequality is $z\ge 6$ .

Since the inequality sign is not strict, use the solid circles at 6 and shade the region to the right side as shown below