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Chapter 3: Problem 104

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

Short Answer

Expert verified
The solution set for \((x-2)^{2}>0\) is all real numbers except \(x=2\), i.e., \(-\infty < x < 2\) or \(2 < x < \infty\).

Step by step solution

01

Recognize Special Form

The inequality is already in a special form, \((x-a)^{2}>0\). Here \(a=2\). The square of a number is always non-negative, meaning it's always greater than or equal to zero. However, because the inequality specifies strictly greater than zero, the squared term cannot equal zero.
02

Find Excluded Values

In our case, the inequality \((x-2)^2 > 0\) equals to zero when \(x=2\). This is the value that needs to be excluded from the solution set.
03

Describe Solution Set

The solution set is all real numbers except for \(x=2\). We can write this as \(-\infty < x < 2\) or \(2 < x < \infty\).

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