Problem 23 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

The solution set is \((-\infty, 1] \cup [3, \infty)\), and the graph is shaded for these intervals.

Step 1: Find the Critical Points

To find the critical points, set each factor of the inequality equal to zero. Solve \((2x - 3) = 0\), \((x - 1)^2 = 0\), and \((x - 3) = 0\). This gives us the critical points: \(x = \frac{3}{2}, x = 1, \text{ and } x = 3\).

Step 2: Determine Test Intervals

The critical points divide the real number line into intervals. The intervals are: \((-\infty, 1)\), \((1, \frac{3}{2})\), \((\frac{3}{2}, 3)\), and \((3, \infty)\).

Step 3: Test Intervals

Choose test points from each interval, such as \(x = 0\), \(x = 1.5\), \(x = 2\), and \(x = 4\), and substitute them into the inequality to test whether the inequality holds in each interval. \((0): (-) \cdot (+) \cdot (-) = +\), \((1.5): (+) \cdot (+) \cdot (-) = -\), \((2): (+) \cdot (+) \cdot (-) = -\), \((4): (+) \cdot (+) \cdot (+) = +\).

Step 4: Include points of zero in the solution

Since \((x-1)^2\) makes the inequality zero at \(x = 1\) (it is a duplicate root, thus contributing to sign change), include \(x = 1\) in the solution. The function is zero at these points: \(x = \frac{3}{2}\) and \(x = 3\). As (2x-3) and (x-3) change sign at \(x=\frac{3}{2}\) and \(x=3\), they are included if the inequality is\(\geq\) sign includes equality.

Step 5: Write the Solution in Interval Notation

The solution set includes all intervals where the inequality is non-negative, which are \((-\infty, 1]\), where it is zero or positive, and \([3, \infty)\). Combine with union: \((-\infty, 1] \cup [3, \infty)\).

Step 6: Graph the Solution

On a number line, shade the interval \((-\infty, 1]\) and \([3, \infty)\). Use a closed dot at \(x = 1\) and \(x = 3\) to indicate that these points are included in the solution set.

Key Concepts

Critical Points

Critical points are the values of \(x\) where the function changes its behavior from increasing to decreasing or vice versa. These points are found by setting each factor in the inequality to zero, essentially `breaking` the inequality into simpler parts. For the inequality \((2x - 3)(x - 1)^2(x - 3) \geq 0\), we identify critical points by solving:

• \((2x - 3) = 0\) simplifies to \(x = \frac{3}{2}\).
• \((x - 1)^2 = 0\) gives us \(x = 1\). This is a duplicate root, indicating it doesn't change sign.
• \((x - 3) = 0\) leads to \(x = 3\).

By identifying these critical points, we know the values of \(x\) that have the potential to change the sign of the entire expression, thus impacting the inequality's solution. Remember, critical points are essential for dissecting and understanding inequalities effectively.

Interval Notation

Interval notation is a mathematical way of describing a set of numbers along an interval of the real number line. It provides a concise way to express where an inequality is true.
Once you have critical points, you use them to divide the number line into intervals. For the resolved inequality \((2x - 3)(x - 1)^2(x - 3) \geq 0\), the intervals are:

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

The solution is expressed using interval notation based on whether the intervals satisfy the inequality. If an interval satisfies, include it; if not, skip it. Don't forget to include closed ends if the inequality is non-strict like \(\geq\).

Sign Analysis

Sign analysis involves checking whether the function is positive or negative across different intervals. This is done by choosing test points from each interval and substituting them back into the original inequality to observe sign behavior.
With our inequality \((2x - 3)(x - 1)^2(x - 3) \geq 0\), we tested using points like

• \(x = 0\) for \(( -\infty, 1)\)
• \(x = 1.5\) for \((1, \frac{3}{2})\)
• \(x = 2\) for \(( \frac{3}{2}, 3)\)
• \(x = 4\) for \((3, \infty)\)

Determine whether each test point results in a positive or negative outcome by observing which intervals satisfy the inequality. Use this data to finalize which intervals to include in your solution.

Graph Sketching

Graph sketching is the visual representation of the solution on a number line, which helps to clearly see where the inequality holds. It's a vital step in confirming your analytical results.
Once the intervals where the inequality holds are identified using sign analysis, represent this on a number line. For the example inequality, shade the intervals \(( -\infty, 1] \) and \( [3, \infty)\). Indicate that these points \(x = 1\) and \(x = 3\) are included in the solution by drawing closed dots at these points.
Creating this graph provides a quick overview of the solution set, ensuring you can easily verify your solution against the inequality criterion.