91Ó°ÊÓ

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. $$\frac{(x-1)^{2}}{x^{2}+4}>0$$

Step by step solution

01

Analyze the Numerator

The numerator of the rational expression is \((x-1)^2\). Since any real number squared is non-negative, \((x-1)^2\) is always greater than or equal to zero. However, the inequality requires that the entire fraction be greater than zero. This means \((x-1)^2\) needs to be strictly positive, excluding the zero case.

02

Consider the Denominator

The denominator is given as \(x^2 + 4\). This expression is always positive because \(x^2\) is non-negative for any real value of \(x\), and adding 4 (a positive number) means \(x^2 + 4\) cannot be zero or negative. Thus, the denominator is strictly positive for all real numbers \(x\).

03

Determine Sign of the Expression

Since the denominator \(x^2 + 4\) is always positive and the fraction must be greater than zero, the numerator \((x-1)^2\) must be positive. \((x-1)^2\) is positive for any \(x eq 1\) because \((x-1)^2 = 0\) only when \(x=1\).

04

Solution Set

The solution to the inequality \(\frac{(x-1)^2}{x^2+4}>0\) thus excludes the point where the numerator is zero. Therefore, the solution set is all real numbers except where \((x-1)^2 = 0\), which occurs at \(x = 1\). Thus, the solution set is \( (-\infty, 1) \cup (1, \infty) \).

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Numerator Analysis

When analyzing rational inequalities, it's crucial to start with the numerator. In our given inequality, \((x-1)^2\) serves as the numerator. The square of any real number is always non-negative, meaning \((x-1)^2\) is ≥ 0. However, because the inequality requires the overall expression to be positive (greater than zero), the numerator cannot equal zero; it must be strictly positive.

• For any value of \(x\) except \(x = 1\), \((x-1)^2\) is positive.
• At \(x = 1\), the numerator becomes zero.

Hence, the numerator analysis reveals that the inequality is satisfied as long as \(x\) does not equal 1. Understanding this part of the expression sets the stage for solving the entire inequality.

Denominator Analysis

Next, let's focus on the denominator, which in our inequality is \(x^2 + 4\). A thorough analysis of this part helps understand its behavior and impact on the inequality.

• Since \(x^2\) is non-negative for any real number, adding 4 makes \(x^2 + 4\) a positive expression for all \(x\).
• There is no real number \(x\) for which this expression could be zero or negative.

This implies the denominator never causes the rational expression to cross through zero or become undefined, making it consistently positive. This is a key observation because it ensures that any change in the sign of the rational expression depends solely on the numerator's sign. By establishing that the denominator is problem-free (always positive), we concentrate on the numerator to identify strict positivity.

Solution Set

With the numerator and denominator fully understood, we can now determine the solution set for the inequality \(\frac{(x-1)^2}{x^2+4} > 0\). The primary goal is to satisfy the condition that the rational expression is positive.

• The denominator \(x^2 + 4\) remains positive everywhere, causing no constraints on the values of \(x\).
• The numerator \((x-1)^2\) must also be positive for the inequality to hold.

Because \((x-1)^2 = 0\) precisely when \(x = 1\), we exclude \(x = 1\) from our solution.
Thus, the solution set encompasses all real numbers except this single point of zero. Therefore, the solution set in interval notation is \((-\infty, 1) \cup (1, \infty)\). This means \(x\) can take any real value except for 1, ensuring the expression remains strictly positive throughout.