Chapter 8, Problem 20PT

To determine

To solve: The inequality $7-\frac{2}{x}<-\frac{5}{x}$ .

Answer to Problem 20PT

The solution of the inequality is $-1<x<0$ .

Explanation of Solution

Given information:

The given inequality is $7+\frac{2}{x}<-\frac{5}{x}$ .

Calculation:

The excluded value for this inequality is 0.

Solve the related equation. The related equation is

  $7+\frac{2}{x}=-\frac{5}{x}$

Eliminate the fraction of the given equation by multiplying each side of the equation by the least common denominator (LCD).

The LCD for the terms is $x$ .

  $\begin{array}{l}7+\frac{2}{x}=-\frac{5}{x}\\ 7\cdot x+\frac{2}{x}\cdot x=-\frac{5}{x}\cdot x\text{ (Multiply by LCD)}\\ 7x+2=-5\text{ (Divide common factors)}\\ 7x=-5-2\text{ (Subtract 2 from both sides)}\\ x=\frac{-7}{7}\text{ (Divide both sides by 7)}\\ x=-1\end{array}$

Draw the vertical lines at the excluded value and at the solution to separate the number lines into interval.

  

Now test the sample value in each interval to determine whether the values in the interval satisfy the inequality.

Test $x=-2$

  $\begin{array}{l}7+\frac{2}{\left(-2\right)}\stackrel{?}{<}-\frac{5}{\left(-2\right)}\\ 7-1\stackrel{?}{<}\frac{5}{2}\\ 6\cancel{<}\frac{5}{2}\end{array}$

Test $x=-0.5$

  $\begin{array}{l}7+\frac{2}{\left(-0.5\right)}\stackrel{?}{<}-\frac{5}{\left(-0.5\right)}\\ 7-4\stackrel{?}{<}10\\ 3<10\end{array}$

Test $x=1$

  $\begin{array}{l}7+\frac{2}{\left(1\right)}\stackrel{?}{<}-\frac{5}{\left(1\right)}\\ 7+2\stackrel{?}{<}-5\\ 9\cancel{<}-5\end{array}$

The solution is true for $x=-0.5$ . Therefore, the solution is $-1<x<0$ .

Hence,

The solution of the inequality is $-1<x<0$ .