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Chapter 2: Problem 45

Solving a Rational Inequality In Exercises \(39-52\) , solve the inequality. Then graph the solution set. $$\frac{2}{x+5}>\frac{1}{x-3}$$

Short Answer

Expert verified
The solution to the inequality is \(x<-5\) OR \(3<x<8\). This can be represented with a number line where the intervals \((-鈭�, -5)\) and \((3, 8)\) are shaded.

Step by step solution

01

Rearrange the inequality

The inequality is rearranged such that one side equals zero, by subtracting \(\frac{1}{x-3}\) from both sides. This gives us: \(\frac{2}{x+5}-\frac{1}{x-3}>0\). This can further be simplified to \(\frac{2x-3-1x-5}{(x+5)(x-3)}>0\), that finally gives \(\frac{x-8}{(x-3)(x+5)}>0\).
02

Find the critical numbers

Set numerator and denominator to zero and solve for \(x\) to find the critical numbers. For numerator, \(x-8=0\) gives \(x=8\). For denominator, \(x+5=0\) gives \(x=-5\) and \(x-3=0\) gives \(x=3\). Thus the critical numbers are \(x=-5\), \(x=3\), and \(x=8\).
03

Determine intervals for testing

Divide the number line into intervals using the critical numbers. The critical numbers divide the number line into four intervals: \((-鈭�, -5)\), \((-5, 3)\), \((3, 8)\), and \((8, 鈭�)\). Test these intervals to determine whether the original inequality is satisfied.
04

Test intervals

Choose a test point from each interval and substitute it into the inequality. If the result is greater than zero, then that interval is part of the solution set, otherwise it is not. After testing the values, we find that the inequality holds true for intervals \((-鈭�, -5)\) and \((3, 8)\).
05

Write the solution set and graph

The solution set is \(x<-5\) OR \(3<x<8\) and is graphed on a number line.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Critical Numbers
Understanding critical numbers is essential for solving rational inequalities like \( \frac{2}{x+5} > \frac{1}{x-3} \). To find critical numbers, we consider points where the expression's value could change from positive to negative or vice versa. These points occur where the numerator equals zero, which creates a zero in the whole expression, or where the denominator equals zero, which creates an undefined expression.

In our exercise, we set the numerator \( x-8 \) and the denominator \( (x-3)(x+5) \) to zero, yielding critical numbers \( x=-5 \), \( x=3 \) and \( x=8 \) after solving. These numbers are crucial in the next steps of the solution, as they help us determine the intervals to test.
Test Intervals in Inequalities
Once critical numbers are obtained, they divide the number line into distinct intervals. In the context of our inequality \( \frac{2}{x+5} > \frac{1}{x-3} \), we have intervals \( (-\infty, -5) \), \( (-5, 3) \), \( (3, 8) \), and \( (8, \infty) \). To determine which intervals satisfy the inequality, we select a test point from each interval and substitute it back into the original inequality.

If the result from the test point is positive, then the interval contributes to the solution. Conversely, if the result is negative, the interval does not satisfy the inequality. It's through this methodical testing process that we can pave a clear path to our inequality's solution.
Graphing Solution Sets
Graphing solution sets is a compelling way to visually present the solutions of inequalities. After determining which intervals satisfy the inequality \( \frac{2}{x+5} > \frac{1}{x-3} \), we can graphically represent the solution set on a number line. The solution to our inequality consists of the intervals \( x<-5 \) OR \( 3
On the number line, we use open circles to mark the critical numbers \( -5 \) and \( 8 \) because these values do not satisfy the inequality. The intervals \( (-\infty, -5) \) and \( (3, 8) \) are graphed using a solid line between the test points, indicating that every number within these ranges is a solution to the inequality. This graphical approach provides a clear visual summary of where the inequality holds true.

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Most popular questions from this chapter

Apple juice has a pH of 2.9 and drinking water has a pH of \(8.0 .\) The hydrogen ion concentration of the apple juice is how many times the concentration of drinking water?

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