Chapter 9.6, Problem 28PPS

To determine

To calculate: The given inequality and check solution.

Answer to Problem 28PPS

the solution is $-4<x<3$ .

Explanation of Solution

Given information: Inequality $\frac{3}{4}-\frac{1}{x-3}>\frac{x}{x+4}$ .

Calculation:

Inequality $\frac{3}{4}-\frac{1}{x-3}>\frac{x}{x+4}$ .

The excluded value for this inequality are $3$ and $-4$ .

Solve the related equation.

  $\frac{3}{4}-\frac{1}{x-3}=\frac{x}{x+4}$

The LCD for the terms is $4\times \left(x-3\right)\times \left(x+4\right)$ .

Multiply both side by LCD

  $\begin{array}{l}3\times \left(x-3\right)\times \left(x+4\right)-4\times \left(x+4\right)=4x\times \left(x-3\right)\\ x^2-11x+52=0\end{array}$

This equation has no real solution because discriminant is negative.

Hence, there are two critical numbers $3$ and $-4$ .

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 $x=-5$

  $\begin{array}{l}\frac{3}{4}-\frac{1}{-5-3}>\frac{-5}{-5+4}\\ \frac{3}{4}+\frac{1}{5+3}>\frac{5}{1}\\ \frac{7}{8}>5\times \end{array}$

Test $x=-3$

  $\begin{array}{l}\frac{3}{4}-\frac{1}{-3-3}>\frac{-3}{-3+4}\\ \frac{3}{4}+\frac{1}{6}>\frac{-3}{1}\\ \frac{22}{24}>-3\end{array}$

Test $x=4$

  $\begin{array}{l}\frac{3}{4}-\frac{1}{4-3}>\frac{4}{4+4}\\ \frac{3}{4}-1>\frac{1}{2}\\ \frac{-1}{4}>\frac{1}{2}\times \end{array}$

The statement is true for $x=-3$ .Therefore, the solution is $-4<x<3$ .