Chapter 7.7, Problem 4ACYP

To determine

To Solve:

  $\sqrt{2x+2}+1\ge 5$

Answer to Problem 4ACYP

Solution region is $x\ge 7$

Explanation of Solution

Given Information:

Given inequality:

  $\sqrt{2x+2}+1\ge 5$

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 $\sqrt{2x+2}+1\ge 5$

  $\begin{array}{l}\text{Since the radicand of a square root must be greater than or equal to zero, }\\ \text{first solve 2x+2}\ge 0\text{ to identify the values of }x\text{ for which the left hand side}\\ \text{of the inequality is defined}\text{.}\\ 2x+2\ge \mathrm{0....................}[\text{set the radicand }\ge \text{ 0]}\\ 2x\ge 0-\mathrm{2....................}[subtract\text{ 2 from each side]}\\ \frac{2x}{2}\ge \frac{-2}{2}\mathrm{........................}[\text{divide each side by 2]}\\ x\ge -1\end{array}$

  $\begin{array}{l}Now,solve\\ \sqrt{2x+2}+1\ge 5\\ \sqrt{2x+2}\ge 5-\mathrm{1.................}[\text{isolate the radical]}\\ \sqrt{2x+2}\ge 4\\ {\left(\sqrt{2x+2}\right)}^2\ge 4^2\mathrm{................}[\text{eliminate the radical]}\\ 2x+2\ge 16\\ 2x\ge 16-\mathrm{2.........................}[subtract\text{ 2 from each side]}\\ 2x\ge 14\\ x\ge \frac{14}{2}\mathrm{.............................}[\text{divide each side by 2]}\\ x\ge 7\end{array}$

  $\begin{array}{l}Combining\\ x\ge -1\\ and\\ x\ge 7\\ \text{The solution region is }x\ge 7,\text{ the intersection of the two regions}\text{.}\\ \text{We can check by taking a value from the region :}\\ x=8\\ \sqrt{2x+2}+1=\sqrt{2(8)+2}+1=\sqrt{18}+1\ge 5\\ So,\text{ the value from the region satisfies the inequality}\text{.}\\ \text{Hence, the solution region is }x\ge 7\end{array}$