Problem 21 - Calculus Early Transcendentals: Pearson New International Edition

Question

Express the solution set of the given inequality in interval notation and sketch its graph. $$ (x+2)(x-1)(x-3)>0 $$

Step-by-Step Solution

Answer

The solution set is \((-\infty, -2) \cup (1, 3) \cup (3, \infty)\).

Step 1: Identify Critical Points

Find the values of \(x\) that make each factor in \((x+2)(x-1)(x-3)>0\) equal to zero. These are \(x+2=0\), \(x-1=0\), and \(x-3=0\), giving critical points \(x = -2\), \(x = 1\), and \(x = 3\).

Step 2: Determine Intervals

The critical points \(-2\), \(1\), and \(3\) divide the number line into the intervals \((-\infty, -2)\), \((-2, 1)\), \((1, 3)\), and \((3, \infty)\).

Step 3: Test Each Interval

Choose a test point in each interval and substitute it into the inequality \((x+2)(x-1)(x-3)>0\). For example, test point \(-3\) in \((-\infty, -2)\) gives a positive product, \(0\) in \((-2, 1)\) gives a negative product, \(2\) in \((1, 3)\) gives a positive product, and \(4\) in \((3, \infty)\) gives a positive product.

Step 4: Determine the Solution Set

The intervals where the product is positive are \((-\infty, -2)\), \((1, 3)\), and \((3, \infty)\). So, in interval notation, the solution is \((-\infty, -2) \cup (1, 3) \cup (3, \infty)\).

Step 5: Sketch the Graph

Draw a number line and mark the critical points at \(-2\), \(1\), and \(3\). Label the intervals where the inequality holds true with a plus sign \((+)\), and shade the corresponding intervals \((-\infty, -2)\), \((1, 3)\), and \((3, \infty)\) to indicate the solution set.

Key Concepts

Critical Points

Critical points in polynomial inequalities like \((x+2)(x-1)(x-3)>0\) are crucial as they determine where the function changes its sign. To find these points, set each factor in the inequality equal to zero: \(x+2=0\), \(x-1=0\), and \(x-3=0\). Solving these gives us the critical points \(x = -2\), \(x = 1\), and \(x = 3\). These points divide the number line into intervals where the inequality can be tested for sign changes. Understanding the role of critical points is essential because they are the boundaries that will help us determine which intervals satisfy the inequality.

Interval Notation

Interval notation is a mathematical shorthand used to describe a set of numbers between two endpoints. In the case of solving inequalities, particularly polynomial inequalities, once we identify the critical points, they help delineate intervals. Each interval can then be tested to see if it satisfies the inequality. In this exercise, the critical points \(-2\), \(1\), and \(3\) divide the number line into these intervals: \((-fty, -2)\), \((-2, 1)\), \((1, 3)\), and \((3, fty)\).
To express these intervals where the inequality holds true, we use union notation: \((-fty, -2) \cup (1, 3) \cup (3, fty)\). This notation communicates concise information about which portions of the number line the solution set includes.

Polynomial Inequalities

Polynomial inequalities involve expressions where one polynomial is set to be greater or less than zero. Solving inequalities like \((x+2)(x-1)(x-3)>0\) involves finding where the entire expression is positive or negative. This requires first identifying critical points where the expression equals zero, dividing the number line into distinct intervals.
In this scenario, analyzing the sign of each interval provides insight into where the inequality holds. You choose a test point from each interval and substitute it back into the inequality. This process determines whether the product of factors is positive or negative, which relates directly to the solution set of the inequality.

Graphing Inequalities

Graphing inequalities helps visually understand which portions of the number line satisfy the inequality conditions. After finding critical points and intervals for the polynomial \((x+2)(x-1)(x-3)>0\), sketch a simple number line graph.
Mark the critical points \(-2\), \(1\), and \(3\) on this line. Then, analyze the sign of each interval by assigning and checking test points as done before. Indicate the intervals where the inequality holds true on your graph by shading them, and use a plus sign \((+)\) to denote positive intervals. This visual representation effectively illustrates the solution set \((-fty, -2) \cup (1, 3) \cup (3, fty)\), providing a comprehensive understanding of the problem's solution.