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Chapter 2: Problem 89

Solve each inequality using a graphing utility. $$\frac{x-4}{x-1} \leq 0$$

Short Answer

Expert verified
The solution to the inequality \( \frac{x-4}{x-1} \leq 0 \) when graphically represented has intervals (-鈭�, 1) and (4, 鈭�) which satisfy the inequality. Note that the function is undefined at x = 1.

Step by step solution

01

Break Down the Inequality

First separate the inequality \( \frac{x-4}{x-1} \leq 0 \) into two parts. Find the roots of each. Set the numerator and the denominator equal to zero to find the critical points. \n For \(x - 4 = 0\), x = 4\n For \(x - 1 = 0\), x = 1. The x values 4 and 1 are the points where the function crosses or touches the x-axis.
02

Identify the Discontinuity

A function is undefined when the denominator is equal to zero, creating a discontinuity. In this case, the denominator \(x - 1 = 0\) when x = 1. So, x = 1 is a point of discontinuity in this function.
03

Find the Solution Intervals

Creating intervals based on the values of x from Step 1 and the discontinuity from Step 2 we have three ranges: (-鈭�, 1), (1, 4), and (4, 鈭�). Substitute a number from each range into the equation to see whether the inequality \( \frac{x-4}{x-1} \leq 0 \) is satisfied.
04

Plot On a Graph and Identify Solution

Each of these intervals correspond to sections on the x-axis and their corresponding y values on the graph. From the graph, identify where the function is less than or equal to 0, and that would be the solution to the inequality. Note the function will be undefined at the point of discontinuity and should be represented by an open circle on the graph.

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