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Chapter 2: Problem 68

Solve each inequality and graph the solution set on a real number line. $$\frac{1}{x+1}>\frac{2}{x-1}$$

Short Answer

Expert verified
The solution to the inequality \( \frac{1}{x+1}>\frac{2}{x-1} \) is \( x < -3 \), with 'x = -1' and 'x = 1' being undefined.

Step by step solution

01

Identify obvious undefined values

First, let's find the values for \(x\) that make the denominator 0 since for these values the expression is undefined. Here we've two cases to consider: \(x = -1\) and \(x = 1\) in denominators render the inequality undefined.
02

Rewrite Inequality without Fractions

Before dealing with fractions, it's easier if we can get rid of them. Multiply each side by \((x+1)(x-1)\), which is the least common denominator (LCD) of the two fractions: \((x+1)(x-1) * \frac{1}{x+1} > (x+1)(x-1) * \frac{2}{x-1}\) Simplifying this gives: \(x - 1 > 2x + 2\)
03

Isolate x on one side

Subtract 'x' and '+2' from both sides to isolate 'x' on one side: \(x - x - 1 - 2 > 2x - x +2 - 2\). This simplifies to: \(-3 > x\) or \(x < -3\)
04

Graph the solution

Plotting the solution on the number line, draw a line and mark '-1', '1', and '-3' as critical points (undefined points and potential solution). The solution is \(x < -3\), mark this part with an open interval on the number line. Note: The solution does not include '1' and '-1' as these two render the original inequality undefined.

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