Chapter 4, Problem 52SGA

To determine

Solve the inequality.

Answer to Problem 52SGA

  $5\le x>54$

Explanation of Solution

Given information:

  $\sqrt{x-5}<7$

Calculation:

Consider the given inequality,

  $\begin{array}{l}\sqrt{x-5}<7squareonbothsides\\ x-5<49\\ x<54\end{array}$

Now $\sqrt{x-5}$ in order to be real number it must be $x-5\ge 0$

  $\begin{array}{l}x\ge 5\\ so,thesolutionis\\ 5\le x<54\end{array}$

Now check this solution by testing values in the intervals defined by the solution. On a number line mark $5and54$ by dashed vertical lines.

Let check the solution by testing values for $x$ in the intervals,

  

  $\begin{array}{l}forx<5,testx=4\\ \sqrt{4-5}<7\\ \sqrt{-1}<7false\end{array}$

  $\begin{array}{l}for5\le x>54,testx=6\\ \sqrt{6-5}<7\\ \sqrt{1}<7,1<7True\\ so5\le x>54isasolution\end{array}$

  $\begin{array}{l}forx\ge 54,testx=55\\ \sqrt{55-5}<7\\ \sqrt{50}<7,7.07\cancel{<}7False\\ sox\ge 54isnotasolution.\end{array}$

Now the solution is graphed on number line,

  

Hence the solution is, $5\le x>54$ .