Chapter 4, Problem 53SGA

To determine

Solve the inequality.

Answer to Problem 53SGA

  $a\ge -1.5$

Explanation of Solution

Given information:

  $4+\sqrt{2a+7}\ge 6$

Calculation:

Consider the given inequality,

  $4+\sqrt{2a+7}\ge 6$

  $\begin{array}{l}\sqrt{2a+7}\ge 2squareonbothsides\\ 2a+7\ge 4\\ 2a\ge -3\\ a\ge -1.5\end{array}$

Now $\sqrt{2a+7}$ in order to be real number it must be $2a+7\ge 0$

  $\begin{array}{l}2a+7\ge 0\\ 2a\ge -7\\ a\ge -3.5\end{array}$

Now check this solution by testing values in the intervals defined by the solution. On a number line mark $-1.5and-3.5$ by dashed vertical lines.

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

  

  $\begin{array}{l}fora<-3.5,testa=-4\\ 4+\sqrt{2\left(-4\right)+7}\ge 6\\ 4+\sqrt{-1}\ge 6false\\ so,a<-3.5,isnotasolution.\end{array}$

  $\begin{array}{l}for-1.5\le x<-3.5,testa=-2\\ 4+\sqrt{2\left(-2\right)+7}\ge 6\\ 4+\sqrt{3}\ge 6\\ 5.732\cancel{\ge }6false\\ so,-1.5\le x<-3.5,,isnotasolution\end{array}$

  $\begin{array}{l}fora\ge -1.5,testa=0\\ 4+\sqrt{2\left(0\right)+7}\ge 6\\ 4+\sqrt{7}\ge 6\\ 6.64\ge 6True\\ so,a\ge -1.5,,isasolution.\end{array}$

Let graphed the solution on number line,

  

Hence the solution is, $a\ge -1.5$ .