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Magazine Chapter 1: Problem 16
Express the solution set of the given inequality in interval notation and sketch its graph. $$ \frac{3 x-2}{x-1} \geq 0 $$
Short Answer
Expert verified
The solution set is \(\left[\frac{2}{3}, 1\right) \cup (1, \infty)\).
Step by step solution
01
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 \(x=1\).
02
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.
03
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 \( \left[ \frac{2}{3}, 1 \right) \cup (1, \infty) \).
04
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 \(\frac{3x-2}{x-1} \geq 0\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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 \( x = 1 \).
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:
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.
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, \( \frac{4}{1} = 4 \).
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:
Using shading, fill in the intervals \( [\frac{2}{3}, 1) \) and \((1, \infty)\):
- **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.
