Chapter 9.6, Problem 8E

To determine

To calculate: The given inequality and check solution.

Answer to Problem 8E

the solution is $-4\le x<1$ .

Explanation of Solution

Given information: Inequality $1+\frac{5}{x-1}\le 0$ .

Calculation:

Inequality $1+\frac{5}{x-1}\le 0$ .

The excluded value for this inequality is $1$ .

Solve the related equation.

  $1+\frac{5}{x-1}=0$

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

Multiply both side by LCD

  $\begin{array}{l}x-1+5=0\\ x+4=0\\ x=-4\end{array}$

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 $x=-5$

  $\begin{array}{l}1+\frac{5}{-5-1}\le 0\\ \frac{1}{6}\le 0\times \end{array}$

Test $x=0$

  $\begin{array}{l}1+\frac{5}{0-1}\le 0\\ -4\le 0\end{array}$

Test $x=2$

  $\begin{array}{l}1+\frac{5}{2-1}\le 0\\ 6\le 0\times \end{array}$

The statement is true for $x=0$ . Therefore, the solution is $-4\le x<1$ .