Chapter 6.7, Problem 58PPS

To determine

To Solve:

  $-2+\sqrt{8-4z}\ge 8$

Answer to Problem 58PPS

Solution region is

  $z\le -23$

Explanation of Solution

Given Information:

Given inequality:

  $-2+\sqrt{8-4z}\ge 8$

Concept Used :

Solve Radical inequalities:

• If the index of the root is even , identify the values of the variable for which the radicand is nonnegative.
• Solve the inequality algebraically.
• Test values to check.

Calculation:

• Given $-2+\sqrt{8-4z}\ge 8$
• $\begin{array}{l}\text{Since the radicand of a square root must be greater than or equal to zero, }\\ \text{first solve 8-4z}\ge 0\text{ to identify the values of }x\text{ for which the left hand side}\\ \text{of the inequality is defined}\text{.}\\ 8-4z\ge \mathrm{0....................}[\text{set the radicand }\ge \text{ 0]}\\ -4z\ge 0-\mathrm{8....................}[subtract\text{ 8 from each side]}\\ \text{4z}\le \text{8}\mathrm{............................}\text{[multiply each side by -1]}\\ \frac{4z}{4}\le \frac{8}{4}\mathrm{.....................}[\text{Divide each side by 4]}\\ z\le 2\end{array}$
• $\begin{array}{l}Now,solve\\ -2+\sqrt{8-4z}\ge 8\\ \sqrt{8-4z}\ge 8+\mathrm{2.................................}[add\text{ 2 each side]}\\ \sqrt{8-4z}\ge 10\\ {\left(\sqrt{8-4z}\right)}^2\ge {\left(10\right)}^2\mathrm{................}[\text{eliminate the radical]}\\ 8-4z\ge 100\\ -4z\ge 100-\mathrm{8.........................}[subtract\text{ 8 from each side]}\\ -4z\ge 92\\ 4z\le -\mathrm{92................................}[\text{multiply each side by -1]}\\ \frac{4z}{4}\le \frac{-92}{4}\mathrm{........................................}[\text{divide 4 each side]}\\ z\le -23\end{array}$
• $\begin{array}{l}Combining\\ z\le -23\\ and\\ z\le 2\\ \text{The solution region is }z\le -23,\text{ the intersection of two regions}\text{.}\end{array}$