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Chapter 3: Problem 107

Use inspection to describe each inequality's solution set. Do not solve any of the inequalities. $$\frac{1}{(x-2)^{2}}>0$$

Short Answer

Expert verified
The solution set for the inequality \( \frac{1}{(x-2)^{2}} > 0 \) is all real numbers except for \(x=2\), where the equation is undefined.

Step by step solution

01

Understand the Properties of the Second Power

In this step, it is noticed that \((x-2)^{2}\) is a square. A square of any real number, whether it's positive or negative, is always positive or, at the least, zero.
02

Identify the Exception in the Solution Set

While \((x-2)^{2}\) is always positive, there exists an exception. When \(x=2\), \((x-2)^{2}\) equals zero, making the denominator equals zero which is undefined in the real numbers. Thus, \(x\) cannot be 2.
03

Combining Observations for the Solution Space

In the final step, based on previous observations, it is known that \((x-2)^{2}\) is positive for any real \(x\) except for when \(x=2\), where it equals zero. Considering we're inspecting the condition where \( \frac{1}{(x-2)^{2}} > 0 \), it means that the solution set will be any real number except 2, since zero is undefined.

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