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Chapter 2: Problem 67

Solve each inequality and graph the solution set on a real number line. $$\frac{3}{x+3}>\frac{3}{x-2}$$

Short Answer

Expert verified
The solution to the inequality is set of all real numbers excluding -3 and 2. So, the solution set is wrote in interval notation as: \((-\infty, -3) \cup (-3, 2) \cup (2, \infty)\).

Step by step solution

01

Simplify the Inequality

First, get rid of the fractions on both sides by cross multiplying. This will make things simpler to deal with. So, the inequality \(\frac{3}{x+3}>\frac{3}{x-2}\) becomes \((x+3) * 3 > (x-2) * 3\), which simplifies to \(3x + 9 > 3x - 6\)
02

Solve the Inequality

Next, try to isolate x in the inequality by subtracting \(3x\) from both sides. This gives \(9 > -6\). This inequality always holds true for every real number. But we must also consider the values that will make the denominator of the original expression zero, which should be excluded.
03

Determine the Excluded Values for x

We have to consider the values that will make the denominator equal to zero which can't be the solution for x. So, we equate \(x+3 = 0\) and \(x-2 = 0\), which gives us \(x = -3\) and \(x = 2\) respectively. They can't be solutions of the given inequality expression and must be excluded.
04

Graph the Solution

The solution set consists of every real number except -3 and 2. On the number line, this will be represented by all points except -3 and 2. Those points should be represented with hollow circles, indicating they're excluded.

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Most popular questions from this chapter

Solve each inequality using a graphing utility. $$\frac{x-4}{x-1} \leq 0$$

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