Chapter 6, Problem 76SGR

To determine

The solution for the inequality $\sqrt{3x+13}-5\ge 5$ .

Answer to Problem 76SGR

The solution for the inequality $\sqrt{3x+13}-5\ge 5$ is $x\in \left[29,\infty \right)$ .

Explanation of Solution

Given information: The given inequality is $\sqrt{3x+13}-5\ge 5$ .

Calculation: The given inequality $\sqrt{3x+13}-5\ge 5$ can be simplified as,

  $\begin{array}{c}\sqrt{3x+13}-5\ge \mathrm{5....}\left(\text{Given}\right)\\ \sqrt{3x+13}\ge 10\\ 3x+13\ge \mathrm{100....}\left(\text{Squared}\right)\\ 3x\ge 87\\ x\ge 29\end{array}$

Also for the square-root function,

  $\begin{array}{c}3x+13\ge 0\\ x\ge -\frac{13}{3}\end{array}$

Hence, the solution for the inequality is

  $\begin{array}{c}x\in \left[29,\infty \right)\cap x\in \left[-\frac{13}{3},\infty \right)\\ x\in \left[29,\infty \right)\end{array}$