Problem 37 - Beginning and Intermediate Algebra

Question

Solve each inequality, and graph the solution set. $$ \frac{x-1}{x-4}>0 $$

Step-by-Step Solution

Answer

The solution set is \((-\infty, 1) \cup (4, \infty)\).

Step 1: Identify Critical Points

To solve the inequality \(\frac{x-1}{x-4}>0\), identify the points where the expression equals zero or is undefined. Set the numerator \(x-1=0\) and the denominator \(x-4=0\) and solve for \(x\). \(x=1\) and \(x=4\) are the critical points.

Step 2: Create Test Intervals

Divide the number line into intervals based on the critical points: \( (-\infty, 1) \), \( (1, 4) \), and \( (4, \infty) \).

Step 3: Test Each Interval

Select a test point from each interval to determine if the inequality is positive in that interval. \((-\infty, 1)\): Pick \(x=0\). \(\frac{0-1}{0-4} = \frac{-1}{-4} > 0\). \( (1, 4) \): Pick \(x=2\). \(\frac{2-1}{2-4} = \frac{1}{-2} < 0\). \( (4, \infty) \): Pick \(x=5\). \(\frac{5-1}{5-4} = \frac{4}{1} > 0\)

Step 4: Determine Solution Set

The inequality \(\frac{x-1}{x-4}>0\) is positive for the intervals where the test point evaluation gave a positive result. Therefore, the solution is the union of the intervals \((-\infty, 1) \cup (4, \infty)\).

Step 5: Graph the Solution Set

On a number line, mark open circles at \(x=1\) and \(x=4\). Shade the intervals \( (-\infty, 1)\) and \((4, \infty)\).

Key Concepts

Critical Points

In solving inequalities like \(\frac{x-1}{x-4}>0\), finding the critical points is crucial. Critical points occur where the expression equals zero or is undefined. For the inequality \(\frac{x-1}{x-4}>0\), we set the numerator \(x-1=0\) and the denominator \(x-4=0\). Solving these gives us the critical points \(x=1\) and \(x=4\).
These points divide the number line into distinct intervals that we need to test to determine where the inequality holds true.

Test Intervals

Once we've identified the critical points (\texttt{x=1} and \texttt{x=4}), the next step is to create test intervals. These intervals are sections of the number line divided by the critical points: \[(-\texttt{\infty},1), (1,4), (\texttt{4, \infty})\].
To determine in which intervals our inequality \(\frac{x-1}{x-4}>0\) holds, we choose a test point from each interval and plug it into the original inequality.

Solution Set Graph

After testing the intervals, we determine where the inequality is positive. In our example, the intervals \((-\texttt{\infty}, 1)\) and \((4, \texttt{\infty})\) are where \(\frac{x-1}{x-4} > 0\).
To graph this solution set, we place open circles at \(x=1\) and \(x=4\) to indicate they are not included in the solution. Then, we shade the intervals that satisfy the inequality. This provides a clear visual representation of the solution set on a number line.