Solve each inequality in Exercises 86–91 using a graphing utility. (x - 4)/(x - 1) ≤ 0

Verified step by step guidance

1. Identify the critical points by setting the numerator and denominator equal to zero separately: solve \(x - 4 = 0\) and \(x - 1 = 0\). These points divide the number line into intervals.
2. Determine the intervals to test based on the critical points found: \(( -\infty, 1 )\), \((1, 4)\), and \((4, \infty)\). Note that \(x = 1\) is excluded because it makes the denominator zero.
3. Choose a test value from each interval and substitute it into the expression \(\frac{x - 4}{x - 1}\) to check whether the expression is positive or negative in that interval.
4. Since the inequality is \(\leq 0\), select the intervals where the expression is negative or zero. Also, include the point where the numerator is zero (\(x = 4\)) because the expression equals zero there.
5. Express the solution set by combining the intervals where the inequality holds, making sure to exclude \(x = 1\) where the expression is undefined.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions where one polynomial is divided by another, and the inequality compares this ratio to zero or another value. Solving them requires finding where the rational expression is positive, negative, or zero, considering the domain restrictions where the denominator is not zero.

Critical Points and Sign Analysis

Critical points occur where the numerator or denominator equals zero, dividing the number line into intervals. By testing values in each interval, you determine the sign of the rational expression, which helps identify where the inequality holds true.

Using Graphing Utilities for Inequalities

Graphing utilities plot the rational function, visually showing where the function is above, below, or equal to zero. This visual aid helps quickly identify solution intervals for the inequality, especially when combined with understanding domain restrictions.