Chapter 5.4, Problem 39E

To determine

To solve : the given inequality.

Answer to Problem 39E

The solution of the given inequality is $x>27$ .

Explanation of Solution

Given information :

The given inequality:

  $4\sqrt{x-2}>20$

Concept used :

To solve a radical inequality of the form $\sqrt[n]{u}<d$ , where $u$ is an algebraic expression and $d$ is a non-negative number, raise each side to the exponent $n$ . This procedure also works for $>,\le ,\text{ and}\ge$ .

Calculation :

As per problem,

  $\begin{array}{l}4\sqrt{x-2}>20\text{ [Write original inequality]}\\ \sqrt{x-2}>5\text{ [Divide each side by 4]}\\ {\left(\sqrt{x-2}\right)}^2>5^2\text{ [Square each side]}\\ x-2>25\text{ [Simplify]}\\ x>27\text{ [Add 2 to each side] }\end{array}$

Now consider the radicand,

  $\begin{array}{l}x-2\ge 0\text{ [The radicand cannot be negative]}\\ x\ge 2\text{ [Add 2 to each side]}\end{array}$

Therefore, the solution of the inequality is $x>27$ .