Chapter 9.6, Problem 26PPS

To determine

To calculate: The given inequality and check solution.

Answer to Problem 26PPS

the solution is $0<a<\frac{11}{10}$ .

Explanation of Solution

Given information: Inequality $\frac{5}{3a}-\frac{3}{4a}>\frac{5}{6}$ .

Calculation:

Inequality $\frac{5}{3a}-\frac{3}{4a}>\frac{5}{6}$ .

The excluded value for this inequality is $0$ .

Solve the related equation.

  $\frac{5}{3a}-\frac{3}{4a}=\frac{5}{6}$

The LCD for the terms is $12a$ .

Multiply both side by LCD

  $\begin{array}{l}20-9=10a\\ 11=10a\\ a=\frac{11}{10}\end{array}$

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 $a=-1$

  $\begin{array}{l}-\frac{5}{3}+\frac{3}{4}>\frac{5}{6}\\ -\frac{11}{12}>\frac{5}{6}\times \end{array}$

Test $a=1$

  $\begin{array}{l}\frac{5}{3}-\frac{3}{4}>\frac{5}{6}\\ \frac{11}{12}>\frac{5}{6}\end{array}$

Test $a=2$

  $\begin{array}{l}\frac{22}{48}>\frac{5}{6}\\ \frac{11}{24}>\frac{5}{6}\times \end{array}$

The statement is true for $a=1$ .Therefore, the solution is $0<a<\frac{11}{10}$ .