Problem 57 - A Graphical Approach to College Algebra

Question

In some cases, it is possible to solve a rational inequality simply by deciding what sign the numerator and the denominator must have and then using the rules for quotients of positive and negative numbers to determine the solution set. For example, consider the rational inequality $$\frac{1}{x^{2}+1}>0$$ The numerator of the rational expression, \(1,\) is positive, and the denominator, \(x^{2}+1,\) must always be positive because it is the sum of a nomegative number, \(x^{2},\) and a positive number, i. Therefore, the rational expression is the quotient of two positive numbers, which is positive. Because the inequality requires that the rational expression be greater than 0 , and this will always be true, the solution set is ( \(-\infty, \infty\) ). Use similar reasoning to solve each inequality. $$\frac{(x-1)^{2}}{x^{2}+4} \leq 0$$

Step-by-Step Solution

Answer

The solution set is \(\{1\}\).

Step 1: Analyze the Numerator

The numerator of the rational expression is \[(x-1)^2\]Since this is a perfect square, it is always non-negative and equals zero when \(x = 1\). Therefore, the numerator will never be negative.

Step 2: Analyze the Denominator

The denominator of the rational expression is \[x^2 + 4\]Since \(x^2\) is always non-negative and \(4\) is positive, this sum is always positive, for all real \(x\).

Step 3: Determine Conditions to Satisfy Inequality

The inequality \[\frac{(x-1)^2}{x^2+4} \leq 0\]is only satisfied when the entire rational expression equals zero because the expression cannot be negative (since both the numerator and denominator are non-negative). This occurs precisely when the numerator is zero, i.e., when \((x-1)^2 = 0\).

Step 4: Solve for Equality

Solve the equation \((x-1)^2 = 0\)This implies \(x-1 = 0\), so \(x = 1\).

Step 5: Construct the Solution Set

Since the rational expression can be zero only when \(x = 1\) and never negative, the solution set is simply \([1, 1]\), or more commonly written as \(\{1\}\), representing the single solution.

Key Concepts

Numerator Analysis

In dealing with rational inequalities, analyzing the numerator is crucial to understand how it influences the inequality. For the expression \((x-1)^2\), we can observe it's a perfect square. This means:

• A perfect square is always non-negative.
• It equals zero at specific points, here at \(x = 1\).

Since the numerator \((x-1)^2\) never becomes negative, it plays a key role in determining whether the rational expression could be zero or positive.
If the goal is to check where the expression could equal zero, focus on the points where the numerator is zero.

Denominator Analysis

Analyzing the denominator helps ensure we understand when the expression is defined and positive. The expression \(x^2 + 4\) is vital here.

• \(x^2\) is always a non-negative term since it's a square.
• Adding 4, a positive number, makes the entire denominator positive for all real \(x\).

Knowing the denominator is always positive means the rational expression will never be undefined. Understanding this allows us to focus on where the expression meets the given inequality.

Inequality Solution

To solve the inequality \(\frac{(x-1)^2}{x^2+4} \leq 0\), we need to find where the rational expression could be zero.

• The expression is zero when the numerator is zero and the denominator is not zero (as it's always positive).
• Since the numerator is zero at \((x-1)^2 = 0\), solve for \(x\) to find \(x = 1\).

The inequality cannot be negative as established, so this solution satisfies \(\leq 0\). Focusing on these zero points is key to finding where the expression meets the condition.

Solution Set

The solution set represents the values of \(x\) that satisfy the inequality. Here, the expression can be zero only at \(x = 1\).

• The analysis shows no other values make the expression non-positive.
• Thus, the solution set is \(\{1\}\), indicating a singular solution.

Understanding the solution set as \(\{1\}\) highlights that in rational inequalities, sometimes the solution is just the instances where the expression equals zero. Recognizing this simplifies solving similar problems.