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Chapter 3: 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

Understanding the Inequality

Firstly, it is important to realize that \((x-2)^{2}\) is the square of a number (in this case, \(x - 2\)). Any square of a number is always either positive or zero.
02

Applying the Non-negativity Property

In the case of the inequality \((x-2)^{2} \leq 0\), the non-negativity property of squares leads us to understand that the left-hand side of the inequality, \((x-2)^{2}\), can only be equal to zero to satisfy the inequality. If it was more than zero, it would not meet the requirements of the \(\leq 0\) inequality.
03

Describing the Solution Set

Since the value of \((x-2)^{2}\) can only be zero, it means \(x - 2 = 0\). Solving for \(x\), we get \(x = 2\). Therefore, the solution set for this inequality is {2}, meaning the only value of \(x\) that makes the inequality true is 2.

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Most popular questions from this chapter

Use a graphing utility to graph \(y=\frac{1}{x^{2}}, y=\frac{1}{x^{4}},\) and \(y=\frac{1}{x^{6}}\) in the same viewing rectangle. For even values of \(n\), how does changing \(n\) affect the graph of \(y=\frac{1}{x^{2}} ?\)

Use the four-step procedure for solving variation problems given on page 424 to solve. \(y\) varies jointly as \(m\) and the square of \(n\) and inversely as \(p\) \(y=15\) when \(m=2, n=1,\) and \(p=6 .\) Find \(y\) when \(m=3, n=4,\) and \(p=10\).

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I've noticed that \(f(-x)\) is used to explore the number of negative real zeros of a polynomial function, as well as to determine whether a function is even, odd, or neither.

Write an equation that expresses each relationship. Then solve the equation for \(y .\) \(x\) varies directly as the cube of \(z\) and inversely as \(y .\)

Determine whether cach statement is true or false If bhe statement is false, make the necessary change(s) to produce a true statement. The graph of a rational function can have three vertical asymptotes.
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