Chapter 8, Problem 21PT

To determine

To calculate: The given inequality and check solution.

Answer to Problem 21PT

the solution is $x\le \frac{1}{2}$ or $x>1$ .

Explanation of Solution

Given information: Inequality $7+\frac{2}{x}<\frac{-5}{x}$ .

Calculation:

Inequality $7+\frac{2}{x}<\frac{-5}{x}$ .

The excluded value for this inequality is $0$ .

Solve the related equation.

  $7+\frac{2}{x}=\frac{-5}{x}$

The LCD for the terms is $x$ .

Multiply both side by LCD

  $\begin{array}{l}7x+2=-5\\ 7x=-7\\ x=-1\end{array}$

Draw vertical lines at the excluded value and at the solutions to separates the number line into intervals.

  

Now test a sample value in each interval to determine whether the values in the interval satisfy the inequality.

Test $x=-2$

  $\begin{array}{l}7+\frac{2}{-2}<\frac{-5}{-2}\\ 7-1<\frac{5}{2}\\ 6<\frac{5}{2}\times \end{array}$

Test $x=\frac{-1}{2}$

  $\begin{array}{l}7-4<10\\ 3<10\end{array}$

Test $x=2$

  $\begin{array}{l}2+1\ge 0\\ 3\ge 0\end{array}$

The statement is true for $x=\frac{-1}{2}$ . Therefore, the solution is $-1<x<0$ .