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Magazine Chapter 8: Problem 16
Solve each inequality. Write the solution set in interval notation. $$\frac{x-5}{x-6}>0$$
Short Answer
Expert verified
The solution is \((-\infty, 5) \cup (6, \infty)\).
Step by step solution
01
Identify Critical Points
To solve the inequality \( \frac{x-5}{x-6} > 0 \), start by identifying the critical points where the expression is either zero or undefined. The expression is zero when the numerator is zero: \( x - 5 = 0 \), solving this gives \( x = 5 \). The expression is undefined when the denominator is zero: \( x - 6 = 0 \), solving this gives \( x = 6 \). So, the critical points are \( x = 5 \) and \( x = 6 \).
02
Determine the Test Intervals
The critical points \( x = 5 \) and \( x = 6 \) divide the number line into three intervals: \( (-\infty, 5) \), \( (5, 6) \), and \( (6, \infty) \). We need to test each interval to see where the inequality \( \frac{x-5}{x-6} > 0 \) holds true.
03
Test the Intervals
Choose a test point from each interval: \( x = 4 \) for \((-\infty, 5)\), \( x = 5.5 \) for \((5, 6)\), and \( x = 7 \) for \((6, \infty)\). Substitute these into \( \frac{x-5}{x-6} \): - For \( x = 4 \), \( \frac{4-5}{4-6} = \frac{-1}{-2} = \frac{1}{2} > 0 \). - For \( x = 5.5 \), \( \frac{5.5-5}{5.5-6} = \frac{0.5}{-0.5} = -1 < 0 \).- For \( x = 7 \), \( \frac{7-5}{7-6} = \frac{2}{1} = 2 > 0 \).
04
Write the Solution in Interval Notation
The inequality \( \frac{x-5}{x-6} > 0 \) is true for the intervals where the test values resulted in a positive number. This happens for the intervals \( (-\infty, 5) \) and \( (6, \infty) \). Therefore, the solution in interval notation is \( (-\infty, 5) \cup (6, \infty) \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Critical Points
When solving inequalities, critical points are essential as they highlight where the expression can either be zero or become undefined. These points are vital because they divide the number line into different regions, allowing us to analyze the behavior of the inequality in each region.
For the inequality \( \frac{x-5}{x-6} > 0 \), we find the critical points by setting the numerator and denominator separately to zero.
For the inequality \( \frac{x-5}{x-6} > 0 \), we find the critical points by setting the numerator and denominator separately to zero.
- Numerator: The expression becomes zero when the numerator \( x-5 \) is zero, leading to the critical point \( x=5 \).
- Denominator: The expression becomes undefined when the denominator \( x-6 \) is zero, giving us another critical point \( x=6 \).
Interval Notation
After determining critical points, the next step is writing the intervals that represent where the inequality holds true. Interval notation is a compact way to express these solutions, and it uses parentheses and brackets to convey points of inclusivity:
- Parentheses \(( \) or \()\): Used if the endpoint is not included in the interval.
- Brackets \([ \) or \()\): Used if the endpoint is included in the interval.
Test Intervals
Testing intervals is a crucial part of solving inequalities. After identifying the critical points, use them to divide the number line into segments or intervals. You then "test" each interval to determine if the inequality is satisfied.
To test an interval, select any number within that segment:
To test an interval, select any number within that segment:
- For \( (-\infty, 5) \), choose \( x = 4 \).
- For \( (5, 6) \), choose \( x = 5.5 \).
- For \( (6, \infty) \), choose \( x = 7 \).
- \( x = 4: \frac{4-5}{4-6} = \frac{1}{2} > 0 \), so this interval satisfies the inequality.
- \( x = 5.5: \frac{5.5-5}{5.5-6} = -1 < 0 \), so this interval does not satisfy the inequality.
- \( x = 7: \frac{7-5}{7-6} = 2 > 0 \), so this interval satisfies the inequality.
