Chapter 6.7, Problem 10E

To determine

Explain how you could apply the technique in the first example to solving an equality.

Explanation of Solution

Calculation:

Given any inequality $2\sqrt{x}>\sqrt{x+2}+1$ , by the method in the example 1, first find the solution of the equation $2\sqrt{x}=\sqrt{x+2}+1$ .

So, the solutions of the equation $\sqrt{x+2}-2\sqrt{x}+1=0$ is :

  

x = 2.398 $\approx 2.4\Rightarrow x>2.4$ , since the sign in the original inequality is > , so the sign will be > in the solution.

Also , since the index is 2 , which is even so , the radican is non-negative , so, $\begin{array}{l}x\ge 0\\ x+2\ge 0\Rightarrow x\ge -2\end{array}$

Combining all the three values of x, we can find the solution .

So, the solution is $x>2.4$ .

In this manner , we can use example 1 to solve the inequality.