Chapter 8.6, Problem 29PPS

To determine

To calculate: The given inequality and check solution.

Answer to Problem 29PPS

the solution is $x<-5$ or $4<x<\frac{17}{3}$ .

Explanation of Solution

Given information: Inequality $\frac{x}{5}+\frac{2}{3}<\frac{3}{x-4}$ .

Calculation:

Inequality $\frac{x}{5}+\frac{2}{3}<\frac{3}{x-4}$ .

The excluded value for this inequality is $4$ .

Solve the related equation.

  $\frac{x}{5}+\frac{2}{3}=\frac{3}{x-4}$

The LCD for the terms is $15\times \left(x-4\right)$ .

Multiply both side by LCD

  $\begin{array}{l}3x\times \left(x-4\right)+10\times \left(x-4\right)=45\\ 3x^2-12x+10x-40=45\\ 3x^2-2x-85=0\\ \left(3x-17\right)\left(x+5\right)=0\\ x=\frac{17}{3},-5\end{array}$

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 $x=-6$

  $\begin{array}{l}\frac{-6}{5}+\frac{2}{3}<\frac{3}{-6-4}\\ \frac{-8}{15}<\frac{-3}{10}\\ \frac{8}{15}>\frac{3}{10}\end{array}$

Test $x=0$

  $\frac{2}{3}<\frac{3}{-4}\times$

Test $x=5$

  $\begin{array}{l}\frac{5}{5}+\frac{2}{3}<\frac{3}{5-4}\\ \frac{5}{3}<3\end{array}$

Test $x=6$

  $\begin{array}{l}\frac{6}{5}+\frac{2}{3}<\frac{3}{6-4}\\ \frac{28}{15}<\frac{3}{2}\times \end{array}$

The statement is true for $x=-6$ and $x=5$ .Therefore, the solution is $x<-5$ or $4<x<\frac{17}{3}$ .