Chapter 6.7, Problem 51PPS

To determine

To Solve:

  $\sqrt{2x+14}-6\ge 4$

Answer to Problem 51PPS

Solution region is $x\ge 43$

Explanation of Solution

Given Information:

Given inequality:

  $\sqrt{2x+14}-6\ge 4$

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 $\sqrt{2x+14}-6\ge 4$

$\begin{array}{l}\text{Since the radicand of a square root must be greater than or equal to zero, }\\ \text{first solve 2x+14}\ge 0\text{ to identify the values of }x\text{ for which the left hand side}\\ \text{of the inequality is defined}\text{.}\\ 2x+14\ge \mathrm{0....................}[\text{set the radicand }\ge \text{ 0]}\\ 2x\ge 0-\mathrm{14....................}[subtract\text{ 14 from each side]}\\ \frac{2x}{2}\ge \frac{-14}{2}\mathrm{........................}[\text{divide each side by 2]}\\ x\ge -7\end{array}$

$\begin{array}{l}Now,solve\\ \sqrt{2x+14}-6\ge 4\\ \sqrt{2x+14}\ge 4+\mathrm{6.................}[\text{isolate the radical]}\\ \sqrt{2x+14}\ge 10\\ {\left(\sqrt{2x+14}\right)}^2\ge {10}^2\mathrm{................}[\text{eliminate the radical]}\\ 2x+14\ge 100\\ 2x\ge 100-\mathrm{14.........................}[subtract\text{ 14 from each side]}\\ 2x\ge 86\\ x\ge \frac{86}{2}\mathrm{.............................}[\text{divide each side by 2]}\\ x\ge 43\end{array}$

$\begin{array}{l}Combining\\ x\ge -7\\ and\\ x\ge 43\\ \text{The solution region is }x\ge 43,\text{ the intersection of the two regions}\text{.}\\ \text{We can check by taking a value from the region :}\\ x=44\\ \sqrt{2x+14}-6=\sqrt{2(44)+14}-6=\sqrt{102}-6\ge 4\\ So,\text{ the value from the region satisfies the inequality}\text{.}\\ \text{Hence, the solution region is }x\ge 43\end{array}$