Chapter 9.6, Problem 9E

To determine

To calculate: The given inequality and check solution.

Answer to Problem 9E

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

Explanation of Solution

Given information: Inequality $2+\frac{1}{x-1}\ge 0$ .

Calculation:

Inequality $2+\frac{1}{x-1}\ge 0$ .

The excluded value for this inequality is $1$ .

Solve the related equation.

  $2+\frac{1}{x-1}=0$

The LCD for the terms is $x-1$ .

Multiply both side by LCD

  $\begin{array}{l}2x-2+1=0\\ 2x-1=0\\ x=\frac{1}{2}\end{array}$

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 $x=0$

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

Test $x=\frac{3}{4}$

  $\begin{array}{l}2+\frac{4}{3-4}\ge 0\\ -2\ge 0\times \end{array}$

Test $x=2$

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

The statement is true for $x=0$ and $x=2$ . Therefore, the solution is $x\le \frac{1}{2}$ or $x>1$ .