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Chapter 2: Problem 46

Solve the inequality. Then graph the solution set. $$\frac{5}{x-6}>\frac{3}{x+2}$$

Short Answer

Expert verified
The solution to the inequality is \(x \in (-\infty,-14] \cup (-2,6)\).

Step by step solution

01

Simplify the Inequality

To simplify the inequality, first find a common denominator for both expressions, which is \((x-6)(x+2)\), and then subtract the two expressions to get 1 inequality. \(\frac{5(x+2) - 3(x-6)}{(x-6)(x+2)}>0\). This simplifies to \(\frac{5x+10-3x+18}{(x-6)(x+2)}>0 \Rightarrow \frac{2x+28}{(x-6)(x+2)}>0\).
02

Identify the critical points

The critical points are the values of x for which the numerator or the denominator is zero. The numerator is 0 for x = -14 and the denominator is 0 for x = 6 and x = -2. Therefore, these are the critical points.
03

Test Intervals

For x < -14, both the numerator and the denominator are negative, so the inequality holds. For -14 < x < -2, the numerator is positive and the denominator is negative, so the inequality does not hold. For -2 < x < 6, both the numerator and denominator are positive, so the inequality holds. For x > 6, the numerator is positive and the denominator is negative, so the inequality doesn't hold.
04

Write down the solution in interval notation

The solution set for values of x where the inequality is true, is the set of all x such that \(x \in (-\infty,-14] \cup (-2,6)\).

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