Problem 105 - Algebra and Trigonometry

Question

In Exercises \(104-107,\) use inspection to describe each inequality's solution set. Do not solve any of the inequalities. $$ (x-2)^{2} \leq 0 $$

Step-by-Step Solution

Answer

The solution set to the inequality \( (x-2)^{2} \leq 0 \) is \( x = 2 \).

Step 1: Understand the inequality

The inequality given here is \( (x-2)^{2} \leq 0 \). When any expression is squared, its minimum value is zero. It can never be less than zero.

Step 2: Define the solution set

Given that \( (x-2)^{2} \leq 0 \), the only value that satisfies this inequality is when \( (x-2)^{2} = 0 \). Solve the equality and one finds \( x = 2 \).

Key Concepts

Solution Set

When dealing with inequalities, one of the primary goals is to determine the solution set. This is essentially the collection of all values that satisfy the given inequality. In our example, the inequality is

• \((x-2)^2 \leq 0\)

Here, we need to determine when the squared expression

• \((x-2)^2\)

equals zero. This approach helps us understand what values of \(x\) make the inequality hold true.
In our exercise, the solution set is simply \(x = 2\), because this is the only value that turns the expression into zero. Thus, the solution set contains just one element.

Squared Expression

A squared expression is any mathematical expression where a variable or number is multiplied by itself. In our case, the expression is

• \((x-2)^2\)

Squared expressions have special properties. They are always non-negative, meaning they are never less than zero.
The key reason behind this is that squaring any real number or expression makes negative values positive, while zero remains unchanged. This is why \\((x-2)^2\)\ever being negative is impossible.
Understanding this property simplifies the solving process of many inequalities, especially when linked with inequality theory.

Non-Negative Values

The concept of non-negative values is crucial when dealing with squared expressions. It refers to values that are either above zero or equal to zero, but not below it.
For the inequality

• \((x-2)^2 \leq 0\)

to hold true, it demands

• \((x-2)^2\)

be equal to zero, which is why the only possible solution is \(x = 2\).
In this situation, it’s particularly important to remember that zero is considered a non-negative number, making non-negative a broader term than positive.
Hence, inequalities involving square terms often either have no solutions or ones that revolve precisely around zero.

Inequality Theory

Inequality theory provides insights on how different algebraic expressions compare in terms of size or value. In the scope of squared expressions,

• \((x-2)^{2} \leq 0\)

deals exclusively with equal-to-zero scenarios, because squared terms are always non-negative.
Inequality theory allows us to conclude that no smaller values exist, as no real number when squared gives a negative outcome.
Thus, baked into this theory is a focus on identifying when equality occurs, which becomes essential in solving various algebraic inequalities.
Understanding this can empower you to solve problems more intuitively, by predicting the behavior of expressions without detailed computation.