Problem 105 - College Algebra

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 to the inequality \((x-2)^{2} \leq 0\) is \(x = 2\) since squares are always positive or zero.

Step 1: Understand the equation

The equation is in the form of \((x-2)^{2}\). By the rule of mathematics, the square of any real number (be it positive or negative) is always positive or zero. That being, the left-hand side of the equation can be zero if and only if the term inside the parentheses i.e. \(x – 2\) equals zero.

Step 2: Determine the solutions

Given the analysis in step 1, and since the inequality includes '\(\leq\)' (which indicates 'equal to' or 'less than'), the term inside the parentheses i.e. \(x – 2\) can be equal to zero. Solving \(x – 2 = 0\) yields \(x = 2\).

Key Concepts

Solution Sets

Understanding solution sets is crucial when working with quadratic inequalities. A solution set is essentially a group of all potential values that fulfill the condition of a given equation or inequality.
In the context of the exercise provided, the inequality \((x-2)^{2} \leq 0\) suggests that the squared term must satisfy this condition.
Let's break it down:

• When dealing with a squared term like \((x-2)^{2}\), remember that it represents a parabolic curve when plotted on a graph.
• This curve will always rest at or above the horizontal axis, given that it squares values, leaving no room for negative results.
• The task is to find the x-values that make the inequality true.
  In this instance, the only situation where \((x-2)^{2}\) is zero is when \(x = 2\).

Thus, the solution set here consists solely of \({x = 2}\). Understanding and identifying solution sets require keenly observing the behavior of the function within the assigned constraint, which is critical in ensuring no potential solution is overlooked.

Mathematical Inspection

Mathematical inspection can be a valuable method for addressing inequalities, allowing students to intuitively determine solution sets without extensive calculations.
In problems like the one provided, inspection helps us assess the given information and readily determine feasibility.
Here's how mathematical inspection unfolds:

• Begin by analyzing the form of the inequality. For \((x-2)^{2} \leq 0\), we note the presence of a perfect square.
• Understand the nature of squaring, which yields non-negative results, fostering the realization that minus values are impossible.
• From inspection, conclude whether zero is achievable and under what circumstances.
  Our inequality proves that zero is attainable through \((x-2)\), emphasizing that \(x-2\) must equate to zero.

No extensive calculations are needed for such observations because the inspection focuses on discerning logical implications directly from the inequality itself.

Inequality Solving Concepts

Solving quadratic inequalities involves multiple concepts that come together to identify potential solutions within the context of the inequality. Understanding these concepts can make approaching such problems more straightforward.
Several key elements aid in solving quadratic inequalities like \((x-2)^{2} \leq 0\):

• The Concept of Non-Negativity: Since squares are always non-negative, the central issue becomes determining when they equal zero, as they cannot be less than zero.
• Identifying Critical Points: Finding values, like \(x=2\), where the expression turns zero helps isolate the solution.
• Testing and Verifying: Since you already know that squares yield non-negative outcomes, you must see where equality—rather than strict inequality—is satisfied.
• Graphical Understanding: Visualizing the parabolic nature of the quadratic function can offer additional insight into the range and limitations of the solution.

By being aware of these facets, students are better equipped to confidently approach and solve quadratic inequalities, weighing all possible factors and reasoning through the solutions logically.