Problem 50 - Precalculus Mathematics for Calculus

Question

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$(x-5)(x-2)(x+1)>0$$

Step-by-Step Solution

Answer

The solution is \((-1, 2) \cup (5, \infty)\).

Step 1: Identify Critical Points

To solve the inequality \((x-5)(x-2)(x+1) > 0\), we first identify the critical points, which are the values of \(x\) that make each factor equal to zero. Thus, set each factor equal to zero: \(x-5=0\), \(x-2=0\), and \(x+1=0\). Solving these gives critical points at \(x=5\), \(x=2\), and \(x=-1\).

Step 2: Divide the Number Line

Divide the number line into intervals based on the critical points: \((-\infty, -1)\), \((-1, 2)\), \((2, 5)\), and \((5, \infty)\). These intervals are where the expression \((x-5)(x-2)(x+1)\) may change its sign.

Step 3: Test Sign in Each Interval

Choose a test point from each interval to determine the sign of the expression:1. Interval \((-\infty, -1)\): Test \(x=-2\). The expression is \((-2-5)(-2-2)(-2+1) < 0\).2. Interval \((-1, 2)\): Test \(x=0\). The expression is \((0-5)(0-2)(0+1) > 0\).3. Interval \((2, 5)\): Test \(x=3\). The expression is \((3-5)(3-2)(3+1) < 0\).4. Interval \((5, \infty)\): Test \(x=6\). The expression is \((6-5)(6-2)(6+1) > 0\).

Step 4: Determine Solution Interval

From step 3, see that the expression is positive in intervals \((-1, 2)\) and \((5, \infty)\). Hence, these are the intervals where the inequality is satisfied.

Step 5: Express in Interval Notation

Express the solution set in interval notation: \((-1, 2) \cup (5, \infty)\).

Step 6: Graph the Solution Set

On the number line, shade the intervals \((-1, 2)\) and \((5, \infty)\). Use open circles at \(x=-1\), \(x=2\), and \(x=5\) to indicate that these points are not included in the solution set.

Key Concepts

Critical Points

When solving nonlinear inequalities such as \((x-5)(x-2)(x+1) > 0\), critical points are the key to understanding where the expression might change its sign.These points are where each factor of the polynomial is equal to zero.Here's how you find them:

• Set each factor in the expression equal to zero.
• Solve these simple equations: \(x-5=0\), \(x-2=0\), \(x+1=0\).
• From these, you derive the critical points: \(x=5\), \(x=2\), and \(x=-1\).

These critical points divide the number line into segments, where the expression may change from positive to negative or vice versa.
Identifying critical points is crucial, as they help you figure out which segments of the number line satisfy your inequality.

Interval Notation

Interval notation is a shorthand used to describe the set of solutions for an inequality.After identifying the critical points, the number line is divided into several intervals, and we need to test the sign of the expression within each one.
This notation captures intervals where the inequality holds true.For example, after testing intervals derived from the critical points in our problem:

• \((-\infty, -1)\), not satisfying the inequality; thus not included.
• \((-1, 2)\), where the inequality is satisfied, included in the solution.
• \((2, 5)\), again not satisfying, thus excluded.
• \((5, \infty)\), satisfying, included in the solution.

These are combined using the union symbol \(\cup\) to get the final solution set: \((-1, 2) \cup (5, \infty)\).This notation helps quickly understand continuous sets of numbers that fulfill the inequality.

Solution Sets

Solution sets are simply the collection of values for \(x\) that satisfy the inequality.In our scenario, the solution set derived from testing different intervals is expressed in interval notation.
The correct expression of the solution set is key.For the given inequality:

• The solution set is \((-1, 2) \cup (5, \infty)\).
• An open circle indicates that endpoints are not included in the solution.

This solution set tells us the exact regions on the number line where the inequality holds true.Understanding solution sets is important because they represent the actual answers to the problem, capturing valid intervals without included endpoints.

Graphical Representation

Graphical representation greatly aids in visualizing solution sets of inequalities.When you graph a solution on the number line, it gives you a clear picture of effective ranges.
The process involves:

• Marking critical points with open circles if they aren’t included, indicating no solutions at these exact points.
• Shading the segments of the number line where the inequality is true.
• In our example, shading occurs over the intervals \((-1, 2)\) and \((5, \infty)\).

Such a visual helps in quickly assessing where values satisfy the inequality by revealing intervals clearly.This method provides an immediate check on your algebraic solution and ensures that it aligns with the set theory approach.