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Chapter 2: Problem 105

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

Short Answer

Expert verified
The solution set of the inequality \( (x-2)^{2} \leq 0 \) is \{2\}.

Step by step solution

01

Recognize the Perfect Square

Recognize \( (x-2)^{2} \) as a perfect square. We know that the square of any real number is always non-negative i.e., it is either positive or zero. Therefore, \( (x-2)^{2} \) must be greater than or equal to zero.
02

Examine the Bounding Condition

Here, we have \( (x-2)^{2} \leq 0 \). Since we know that \( (x-2)^{2} \) is always non-negative, it being less than or equal to zero can only be possible when it is equal to zero.
03

Identify the Solution Set

With the above understanding, the solution set for the inequality \( (x-2)^{2} \leq 0 \), can only be when \( (x-2)^{2} = 0 \). This is only possible when \( x = 2 \). So, the solution set is \{2\}.

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