Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

Express the solution set of the given inequality in interval notation and sketch its graph. $$ \frac{1}{3 x-2} \leq 4 $$

Step-by-Step Solution

Answer

The solution set is \((-\infty, \frac{2}{3}) \cup [\frac{3}{4}, \infty)\).

Step 1: Write the Inequality

Start by writing the inequality: \( \frac{1}{3x-2} \leq 4 \). We need to find the values of \( x \) that satisfy this inequality.

Step 2: Reorganize the Inequality

To isolate the fraction, we first subtract 4 from both sides: \( \frac{1}{3x-2} - 4 \leq 0 \). We can express the left side with a common denominator: \( \frac{1 - 4(3x-2)}{3x-2} \leq 0 \), which simplifies to: \( \frac{1 - 12x + 8}{3x-2} \leq 0 \).

Step 3: Simplify the Inequality

Simplify the expression: \( \frac{9 - 12x}{3x-2} \leq 0 \). This will help us find the critical values where the expression equals zero or is undefined.

Step 4: Identify Critical Points

Set the numerator to zero: \( 9 - 12x = 0 \). Solving for \( x \) gives \( x = \frac{3}{4} \). Set the denominator to zero: \( 3x - 2 = 0 \). Solving for \( x \) gives \( x = \frac{2}{3} \). These critical points divide the number line into intervals.

Step 5: Test Intervals

Test the intervals formed by the critical points \( x = \frac{2}{3} \) and \( x = \frac{3}{4} \). Choose test values from the intervals: on the interval \( (-\infty, \frac{2}{3}) \), test \( x = 0 \); on \( (\frac{2}{3}, \frac{3}{4}) \), test \( x = \frac{5}{6} \); and on \( (\frac{3}{4}, \infty) \), test \( x = 1 \).

Step 6: Evaluate Test Points

For each test point, check if \( \frac{9 - 12x}{3x-2} \leq 0 \):- For \( x = 0 \), \( \frac{9 - 12(0)}{3(0) - 2} = -\frac{9}{2} \) (Negative).- For \( x = \frac{5}{6} \), \( \frac{9 - 12(\frac{5}{6})}{3(\frac{5}{6}) - 2} > 0 \)(Positive).- For \( x = 1 \), \( \frac{9 - 12(1)}{3(1) - 2} < 0 \)(Negative).

Step 7: Determine Interval of Solution

The inequality \( \frac{9 - 12x}{3x-2} \leq 0 \) holds for intervals where the expression is negative or zero. Thus, solution intervals are \( (-\infty, \frac{2}{3}) \cup (\frac{3}{4}, \infty) \). Since \( x = \frac{3}{4} \) makes the expression zero, include it.

Step 8: Express Solution in Interval Notation

The solution set is \[ (-\infty, \frac{2}{3}) \cup [\frac{3}{4}, \infty) \].

Step 9: Sketch the Graph

To sketch the solution on a number line: - Use an open circle at \( \frac{2}{3} \) (since it's not included in the solution set).- Use a closed circle at \( \frac{3}{4} \) (since it's included).- Shade the intervals \( (-\infty, \frac{2}{3}) \) and \( [\frac{3}{4}, \infty) \).

Key Concepts

Interval Notation

Interval notation is a mathematical shorthand to describe the set of solutions to an inequality. It efficiently captures all the values that a variable can take without listing each one individually. In our example with the inequality \( \frac{1}{3x-2} \leq 4 \), we found that the solution set is \( (-\infty, \frac{2}{3}) \cup [\frac{3}{4}, \infty) \). These intervals tell us that:

• From negative infinity up to (but not including) \( \frac{2}{3} \), the inequality holds true.
• There is a pause between \( \frac{2}{3} \) and \( \frac{3}{4} \), where the inequality does not hold.
• Then, starting from \( \frac{3}{4} \) (inclusive), extending to positive infinity, the inequality is again true.

Here, parentheses \(( )\) indicate that the endpoint is not included, while square brackets \([ ]\) show that the endpoint is part of the solution. This efficient notation helps avoid long lists of values and makes it easier to understand the set of possible solutions quickly.

Critical Points

Critical points play a crucial role in solving inequalities. They are points on the number line where the inequality changes its sign. For a rational inequality like \( \frac{9 - 12x}{3x-2} \leq 0 \), critical points arise from setting both the numerator and the denominator equal to zero.

• When the numerator of the fraction is zero, the entire fraction equals zero. In our case, \( 9 - 12x = 0 \) leads to the critical point \( x = \frac{3}{4} \).
• When the denominator is zero, the expression is undefined. Here, \( 3x - 2 = 0 \) gives us \( x = \frac{2}{3} \).

The critical points help break down the number line into intervals. These are essential for testing where the inequality holds true, helping understand where the rational expression changes behavior.

Number Line Graph

Graphing solutions on a number line provides a visual representation of which segments satisfy an inequality. In equations like \( \frac{1}{3x-2} \leq 4 \), the number line graph distinctly shows where the inequality holds:

• Place an open circle at \( \frac{2}{3} \) to indicate that it is not part of the solution. The inequality is undefined at this point.
• Put a closed circle at \( \frac{3}{4} \) because the inequality equals zero here, meaning it's included in the solution.

To complete the graph, shade the regions \( (-\infty, \frac{2}{3}) \) and \( [\frac{3}{4}, \infty) \). These shaded areas show the continuous ranges of \( x \) values that make the initial inequality true. The graph serves as a quick reference guiding you through interval implications.

Rational Inequalities

A rational inequality involves quotients of polynomials, like \( \frac{1}{3x-2} \leq 4 \). These inequalities can be tricky due to the presence of variables in the denominators causing fractions to become undefined. To solve such inequalities, follow these steps:

• Reorganize the inequality so one side is zero, as done with \( \frac{1}{3x-2} - 4 \).
• Find critical points by making the numerator zero or the denominator zero, leading to potential changes in the inequality's sign.
• Use test points in different intervals established by the critical points to check where the inequality holds.

Understanding rational inequalities helps manage expressions that behave differently based on the input values. It is important to handle them cautiously due to potential undefined points or jumps in behavior at critical points.