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Chapter 11: Problem 16

Solve each inequality. Write the solution set in interval notation. See Example 4. $$ \frac{x-5}{x-6}>0 $$

Short Answer

Expert verified
\((-\infty, 5) \cup (6, \infty)\)

Step by step solution

01

Find Critical Points

Start by identifying the values of \( x \) that make the expression \( \frac{x-5}{x-6} = 0 \) or undefined. The expression equals zero when the numerator is zero. So, set \( x-5=0 \) and solve for \( x \): \( x=5 \). The expression is undefined when the denominator is zero. So, set \( x-6=0 \) and solve for \( x \): \( x=6 \). Thus, the critical points are \( x=5 \) and \( x=6 \).
02

Number Line and Intervals

Plot the critical points, 5 and 6, on a number line. These points divide the number line into three intervals: \((-\infty, 5)\), \((5, 6)\), and \((6, \infty)\).
03

Test Intervals

Choose a test point from each interval to determine if the inequality is satisfied.- For \((-\infty, 5)\), use \(x=0\): \( \frac{0-5}{0-6} = \frac{-5}{-6} > 0\).- For \((5, 6)\), use \(x=5.5\): \( \frac{5.5-5}{5.5-6} = \frac{0.5}{-0.5} < 0\).- For \((6, \infty)\), use \(x=7\): \( \frac{7-5}{7-6} = \frac{2}{1} > 0\).
04

Combine Results

The inequality \( \frac{x-5}{x-6}>0 \) is satisfied in the intervals where the result is positive. Based on our tests, these intervals are \((-\infty, 5)\) and \((6, \infty)\).
05

Write Solution in Interval Notation

The solution set in interval notation includes both intervals where the expression is positive. So, the solution 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 like \( \frac{x-5}{x-6}>0 \), the first step is to find the critical points. These are values of \( x \) that make the expression either zero or undefined. This gives us crucial information about where the expression might change sign.
Let's break this down:
  • Zero: The expression equals zero when its numerator is zero. Here, setting \( x-5=0 \) gives \( x=5 \).
  • Undefined: The expression is undefined when its denominator is zero. Setting \( x-6=0 \) results in \( x=6 \).
With these calculations, our critical points are \( x=5 \) and \( x=6 \). These points help divide the number line into separate intervals that we will test next.
Interval Notation
Once we have the critical points, we can use them to write intervals. With \( x=5 \) and \( x=6 \), the number line is divided into three intervals:
  • \(( -\infty, 5)\)
  • \((5, 6)\)
  • \((6, \infty)\)
In interval notation, we use parentheses \(()\) to show that endpoints are not included in the interval. This is because the expression \( \frac{x-5}{x-6} \) is not satisfied when exactly at the critical points. We use intervals to express parts of the solution set where our inequality holds true.
This notation is a compact and efficient way to communicate the solution space on a number line.
Test Intervals
With the intervals established, we choose a test point from each to determine where the inequality is true. Testing in each interval helps us decide if the expression \( \frac{x-5}{x-6} > 0 \) holds.
  • \(( -\infty, 5)\): Choose \( x = 0 \). Substituting gives \( \frac{0-5}{0-6}= \frac{-5}{-6} > 0 \), confirming that this interval satisfies the inequality.
  • \((5, 6)\): Choose \( x = 5.5 \). Here, \( \frac{5.5-5}{5.5-6}= \frac{0.5}{-0.5} < 0 \), so this interval does not satisfy the inequality.
  • \((6, \infty)\): Choose \( x = 7 \). Calculations give \( \frac{7-5}{7-6}=\frac{2}{1} > 0 \), so this interval satisfies the inequality.
The intervals that give positive values are the solution, leading to the final conclusion of \(( -\infty, 5) \cup (6, \infty)\). Testing provides clarity on which intervals to include in the solution.

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