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Chapter 3: Problem 51

Solve each rational inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$ \frac{x}{x-3}>0 $$

Short Answer

Expert verified
The solution set of the inequality \( \frac{x}{x-3} > 0 \) in interval notation is (-Infinity, 0) 鈭� (3, +Infinity).

Step by step solution

01

Find the Critical Points

A critical number of the function \( \frac{x}{x-3} \) is a number at which the function either is zero or undefined. Here \( x = 0 \) makes the fraction zero and \( x = 3 \) makes it undefined. Therefore, the critical numbers are 0 and 3.
02

Test Each Region

The critical numbers divide the number line into three regions: (-Infinity, 0), (0, 3), and (3, +Infinity). Now, find a test number for each region and substitute that number into the inequality \( \frac{x}{x-3} > 0 \). The sign of the resultant will tell whether the inequality holds for that region or not. If \( x = -1 \), the result is positive; if \( x = 2 \), it's negative; and if \( x = 4 \), it's positive. So the inequality is true when \(x < 0\) or \( x > 3\).
03

Express the Solution in Interval Notation

The final step is to express the solution set in interval notation. For \( x < 0 \), it is (-Infinity, 0). For \( x > 3 \), it is (3, +Infinity).

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Key Concepts

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

Critical Points
When working with rational inequalities, understanding critical points is crucial. Critical points, also known as critical numbers, are values of the variable that make the expression zero or undefined. In the inequality \( \frac{x}{x - 3} > 0 \), the critical points are found by setting the numerator and denominator to zero.
  • Set the numerator \( x \) to zero, yielding \( x = 0 \). This point makes the expression zero.
  • Set the denominator \( x-3 \) to zero, resulting in \( x = 3 \). This point makes the expression undefined.
Understanding these points helps divide the number line into intervals for sign analysis.
Interval Notation
Interval notation is a method used to denote the set of solutions for inequalities. It provides a compact way to express portions of the number line. In our exercise, the solution to \( \frac{x}{x-3} > 0 \) involves expressing intervals where the inequality holds true. When solving, learn the basics of interval notation:
  • Parentheses \((a, b)\) indicate that endpoints \(a\) and \(b\) are not included in the interval.
  • Brackets \([a, b]\) indicate that the endpoints are included in the interval.
  • Infinity \(\pm \infty\) is always accompanied by parentheses because it is not a specific value.
For this problem, the intervals \((-\infty, 0)\) and \((3, +\infty)\) convey that the inequality holds for \(x < 0\) and \(x > 3\).
Sign Analysis
Sign analysis is essential in determining where an inequality is true or false. By splitting the number line based on critical points, one can test various regions to find where the inequality satisfies the condition given. In the example \( \frac{x}{x-3} > 0 \), sign analysis helps us determine valid intervals. Here's how sign analysis works for this function:
  • Divide the number line according to the critical points \( x = 0 \) and \( x = 3 \), which create intervals \((-\infty, 0)\), \((0, 3)\), and \((3, +\infty)\).
  • Choose a test point in each interval. For this inequality, select \( x = -1 \) for \((-\infty, 0)\), \( x = 2 \) for \((0, 3)\), and \( x = 4 \) for \((3, +\infty)\).
  • Substitute these test points back into the inequality. If the result is positive, the inequality holds true for that interval.
The tests show that \( \frac{x}{x-3} > 0 \) is true for \( x < 0 \) and \( x > 3 \), matching the solution expressed in interval notation.

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