Chapter 7, Problem 81SGR

To determine

The solution for the inequality $\sqrt{3x+1}-\sqrt{6+x}>0$ .

Answer to Problem 81SGR

The solution for the inequality $\sqrt{3x+1}-\sqrt{6+x}>0$ is $x\in \left(\frac{5}{2},\infty \right)$ .

Explanation of Solution

Given information: The given inequality is $\sqrt{3x+1}-\sqrt{6+x}>0$ .

Calculation: The given inequality $\sqrt{3x+1}-\sqrt{6+x}>0$ can be simplified as,

  $\begin{array}{c}\sqrt{3x+1}-\sqrt{6+x}>\mathrm{0....}\left(\text{Given}\right)\\ \sqrt{3x+1}>\sqrt{6+x}\\ 3x+1>6+x\mathrm{....}\left(\text{Squared}\right)\\ 2x>5\\ x>\frac{5}{2}\end{array}$

Hence, the solution for the inequality is

  $x\in \left(\frac{5}{2},\infty \right)$