Problem 16 - Calculus Early Transcendentals: Pearson New International Edition

Question

Express the solution set of the given inequality in interval notation and sketch its graph. $$ \frac{3 x-2}{x-1} \geq 0 $$

Step-by-Step Solution

Answer

The solution set is \(\left[\frac{2}{3}, 1\right) \cup (1, \infty)\).

Step 1: Identify the critical points

To solve the inequality \(\frac{3x-2}{x-1} \geq 0\), we first need to find the values of \(x\) where the fraction is zero or undefined. These critical points will occur when the numerator \(3x-2=0\) or the denominator \(x-1=0\). Solving \(3x-2=0\) gives \(x=\frac{2}{3}\), and solving \(x-1=0\) gives \(x=1\). Hence, the critical points are \(x=\frac{2}{3}\) and \(x=1\).

Step 2: Determine the sign intervals

Next, we determine where the expression \(\frac{3x-2}{x-1}\) is positive, zero, or negative. We consider three intervals created by the critical points: \(x < \frac{2}{3}\), \(\frac{2}{3} < x < 1\), and \(x > 1\). For each interval, pick a test point:- For \(x < \frac{2}{3}\), pick \(x = 0\): \(\frac{3(0)-2}{0-1} = -2\), which is negative.- For \(\frac{2}{3} < x < 1\), pick \(x = 0.8\): \(\frac{3(0.8)-2}{0.8-1} = 1\), which is positive.- For \(x > 1\), pick \(x = 2\): \(\frac{3(2)-2}{2-1} = 4\), which is positive.

Step 3: Construct the solution set

Since the inequality is \(\frac{3x-2}{x-1} \geq 0\), we include intervals where the expression is zero or positive. The expression is zero at \(x = \frac{2}{3}\) and positive on the intervals \(\frac{2}{3} < x < 1\) and \(x > 1\). Since \(x = 1\) makes the expression undefined, we exclude it from the solution set. Thus, the solution set in interval notation is \( \left[ \frac{2}{3}, 1 \right) \cup (1, \infty) \).

Step 4: Sketch the graph

On a number line, mark the points \(\frac{2}{3}\) and \(1\). Use a closed circle at \(\frac{2}{3}\) to represent inclusion in the solution set because the expression is zero there, and an open circle at \(1\) to represent exclusion. Shade the line between \(\frac{2}{3}\) and \(1\), and also shade the line extending from \(1\) to infinity, representing the positive intervals. This visualizes the solution to the inequality \(\frac{3x-2}{x-1} \geq 0\).

Key Concepts

Critical Points

In solving inequalities, identifying critical points is a foundational step. These points are where the expression is equal to zero or becomes undefined. For our inequality \( \frac{3x-2}{x-1} \geq 0 \), critical points occur when the numerator and denominator indicate potential changes in sign.

Here's how we do it:

• Set the numerator equal to zero: \( 3x - 2 = 0 \). Solving gives \( x = \frac{2}{3} \).
• Set the denominator equal to zero: \( x - 1 = 0 \). Solving gives \( x = 1 \).

Therefore, the critical points are \( x = \frac{2}{3} \) and \( x = 1 \). Recognizing these points helps us determine intervals where the expression's sign might change.

Interval Notation

Interval notation is a concise way to express a solution set. It uses parentheses \(()\) and brackets \([]\) to describe intervals on the number line where an inequality holds true. Parentheses indicate the endpoint is not included, while brackets indicate it is.

For the inequality \( \frac{3x-2}{x-1} \geq 0 \), we found that the expression is zero at \( x = \frac{2}{3} \) and undefined at \( x = 1 \). The solution in interval notation thus includes:

• \( [\frac{2}{3}, 1) \): this means \( x \) can range from \( \frac{2}{3} \) to just below \( 1 \), including \( \frac{2}{3} \).
• \( (1, \infty) \): this means that \( x \) is from just above \( 1 \) to infinity, \( 1 \) is not included due to undefined nature.

The complete solution set in interval notation is \([\frac{2}{3}, 1) \cup (1, \infty)\). This tells us where the inequality holds on the number line, allowing for easy visualization.

Sign Intervals

Understanding sign intervals helps determine where a given expression is positive, negative, or zero. This is crucial in solutions for inequalities. For \( \frac{3x-2}{x-1} \), divide the number line into regions based on critical points: \( x < \frac{2}{3} \), \( \frac{2}{3} < x < 1 \), and \( x > 1 \). For each:

• **For \( x < \frac{2}{3} \)**, choose \( x = 0 \). The expression is negative, \( \frac{-2}{-1} = -2 \).
• **For \( \frac{2}{3} < x < 1 \)**, choose \( x = 0.8 \). The expression is positive, \( \frac{1.4}{-0.2} = 1 \).
• **For \( x > 1 \)**, choose \( x = 2 \). The expression is positive, \( \frac{4}{1} = 4 \).

These evaluations indicate where the expression is non-negative, guiding us in forming our solution interval.

Graphical Representation of Inequalities

Graphically representing inequalities can simplify understanding. A number line is perfect for illustrating where an inequality is true. For \( \frac{3x-2}{x-1} \geq 0 \), consider these steps:

• **Mark critical points**: Plot \( x = \frac{2}{3} \) with a closed circle to include it, as the expression equals zero there.
• **Open circle at \( x = 1 \)**: Place an open circle to represent that this point is not part of the solution due to it creating an undefined expression.

Using shading, fill in the intervals \( [\frac{2}{3}, 1) \) and \((1, \infty)\):

• **Shade from \( \frac{2}{3} \) to \( 1 \)**, not including \( 1 \).
• **Continuously shade from \( 1 \) onward**, noting there is no upper bound.

This pictorial helps us to clearly see where the inequality holds true and visually checks our interval notation.