Question

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

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

Step-by-step solution

Identify Critical Points

First, identify the critical points where the expression is either zero or undefined. These occur when the numerator is zero or the denominator is zero. For the numerator \( 3x - 2 = 0 \), solving gives \( x = \frac{2}{3} \). For the denominator \( x - 1 = 0 \), solving gives \( x = 1 \).

Set Up Test Intervals

Using the critical points found, divide the number line into test intervals. Our critical points are \( x = \frac{2}{3} \) and \( x = 1 \), creating the intervals \(( -\infty, \frac{2}{3} ), ( \frac{2}{3}, 1 ), \text{and} (1, \infty) \).

Test Signs in Each Interval

For each interval, pick a test point and determine if the expression \( \frac{3x-2}{x-1} \) is positive or negative. For \(( -\infty, \frac{2}{3} )\), use \( x = 0 \): \( \frac{-2}{-1} > 0 \). For \(( \frac{2}{3}, 1 )\), use \( x = 0.8 \): \( \frac{3(0.8) - 2}{0.8 - 1} < 0 \). For \((1, \infty) \), use \( x = 2 \): \( \frac{6-2}{2-1} > 0 \).

Include Endpoints Appropriately

The inequality \( \geq 0 \) includes the points where the expression equals zero. At \( x = \frac{2}{3} \), the expression is zero, include \( x = \frac{2}{3} \) in the solution set, represented as \[ \left[ \frac{2}{3}, 1 \right) \cup (1, \infty) \].

Sketch the Graph

On a number line, represent the solution set \( \left[ \frac{2}{3}, 1 \right) \cup (1, \infty) \). Draw a filled dot at \( x = \frac{2}{3} \), an open dot at \( x = 1 \), and highlight the intervals \( x \geq \frac{2}{3} \) up to 1 (not including \( x = 1 \)), then from \( x = 1 \) to infinity.

Key concepts

Critical Points

In solving inequalities such as \( \frac{3x-2}{x-1} \geq 0 \), understanding the concept of critical points is crucial. Critical points are values of \( x \) that make the expression either zero or undefined. In this case:

• The numerator \( 3x - 2 = 0 \) results in a critical point at \( x = \frac{2}{3} \).
• The denominator \( x - 1 = 0 \) results in a critical point at \( x = 1 \).

Identifying these points is essential because they help break the number line into different intervals that can be assessed separately. Each interval will tell us whether the expression is positive or negative. To find critical points, set each part of the rational expression to zero and solve for \( x \). The solution of the numerator being zero gives a point where the expression will equal zero in value, while the denominator being zero gives points where the expression is undefined and potentially leads to discontinuities.

Interval Notation

Interval notation is a way of describing a set of numbers along a number line. It is particularly useful in expressing the sets of solutions for inequalities. Once we identify our critical points, we use them to set up our intervals:

• \(( -\infty, \frac{2}{3} )\)
• \(( \frac{2}{3}, 1 ) \)
• \(( 1, \infty )\)

These intervals come from the regions created by our critical points, \( x = \frac{2}{3} \) and \( x = 1 \). In interval notation:

• The symbol \(( \) or \() \) denotes that an endpoint is not included in the interval.
• The symbol \([ \) or \() \) denotes that an endpoint is included.

For our solution to the inequality \( \frac{3x-2}{x-1} \geq 0 \), the interval notation would be \( \left[ \frac{2}{3}, 1 \right) \cup (1, \infty) \). Notice how \( \frac{2}{3} \) is included as the expression equals zero at this point, matching the \( \geq \) sign in the inequality. \( x = 1 \) is not included because it makes the expression undefined.

Inequality Graph

Graphing an inequality like \( \frac{3x-2}{x-1} \geq 0 \) provides a visual representation of the solution set on a number line. Here's how to graph this inequality:

• Plot the critical points \( x = \frac{2}{3} \) and \( x = 1 \) on the number line.
• Use a filled dot at \( x = \frac{2}{3} \) to represent that this point is included in the solution set.
• Use an open dot at \( x = 1 \) to represent it is not included.
• Shade the region from \( x = \frac{2}{3} \) to \( x = 1 \), indicating \( x \geq \frac{2}{3} \) but not including \( x = 1 \).
• Continue shading from \( x = 1 \) to infinity, showing that any \( x \) beyond 1 satisfies the inequality.

Graphing inequalities helps in visualizing the intervals where the inequalities are satisfied. The shaded regions on the graph represent where the expression is either zero or positive, in line with the \( \geq 0 \) condition of our inequality. The correct and clear presentation of filled and open dots helps in categorically distinguishing between included and excluded endpoints. This visual representation makes it easier to understand and verify the solutions derived through algebraic means.