Chapter 2.5, Problem 27E

To determine

To find: Thesolution set for the inequalities $-y+3\le 8$ and $y+2>9$ then also graph it if possible.

Answer to Problem 27E

The solution setofinequalities $-y+3\le 8$ and $y+2>9$ is $y>7$ .

Explanation of Solution

Given:

The inequalities are $-y+3\le 8$ and $y+2>9$ .

Calculation:

The given inequalities are $-y+3\le 8$ and $y+2>9$ .

Solve the first inequality $-y+3\le 8$ .

  $\begin{array}{c}-y+3\le 8\\ -y+3-3\le 8-3\left(\text{add 3 on both sides}\right)\\ -\left(-y\right)\le -\left(5\right)\left(\text{multiply both sides by }-1\right)\\ y\ge -5\left(\text{inequality changed}\right)\end{array}$

Therefore, the solution set for the inequality $-y+3\le 8$ is $y\ge -5$ .

Now, solve the second inequality $y+2>9$ .

  $\begin{array}{c}y+2>9\\ y+2-2\le 9-2\left(\text{subtract 2 on both sides}\right)\\ y>7\end{array}$

Therefore, the solution set for the inequality $y+2>9$ is $y>7$ .

It is mentioned in the question, the word “and” that means the intersection.

Therefore, the intersection of the $y>7$ and $y\ge -5$ is $y>7$ .

Graph the solution set of the given inequalities is shown as Figure 1.

Figure 1

The solution set is dark shaded line thatstarts from the point $7$ and continues to right-side.

Conclusion:

Thus, the solution setofinequalities $-y+3\le 8$ and $y+2>9$ is $y>7$ .