Problem 35 - Precalculus Mathematics for Calculus

Question

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$(x+2)(x-3)<0$$

Step-by-Step Solution

Answer

The solution is \((-2, 3)\) in interval notation.

Step 1: Identify Zero Points

The inequality \[(x+2)(x-3)<0\]will be zero at the points where each factor equals zero. So, solve for:1. \(x+2=0\) gives \(x=-2\).2. \(x-3=0\) gives \(x=3\).These points \(x=-2\) and \(x=3\) divide the number line into intervals.

Step 2: Determine Intervals

Based on the zero points found in Step 1, the number line is divided into the intervals:1. \((-\infty, -2)\)2. \((-2, 3)\)3. \((3, \infty)\)

Step 3: Test Each Interval

Pick a test point from each interval to determine whether the product \[(x+2)(x-3)\]is negative or positive:1. Interval \((-\infty, -2)\): Select \(x = -3\). \((x+2)(x-3) = (-3+2)(-3-3) = (-1)(-6) = 6\) is positive.2. Interval \((-2, 3)\): Select \(x = 0\). \((x+2)(x-3) = (0+2)(0-3) = (2)(-3) = -6\) is negative.3. Interval \((3, \infty)\): Select \(x = 4\). \((x+2)(x-3) = (4+2)(4-3) = (6)(1) = 6\) is positive.

Step 4: Identify Solution Interval

From Step 3, the product is negative in the interval \((-2, 3)\). Thus the inequality \[(x+2)(x-3)<0\]is satisfied for \(x\) in \((-2, 3)\). Because the inequality is strict \(<\), we do not include the endpoints.Therefore, the solution in interval notation is \[(-2, 3)\].

Step 5: Graph Solution Set

To graph the solution set, draw a number line. Mark \(-2\) and \(3\) with open circles to indicate that these points are not included in the solution set. Shade the region between \(-2\) and \(3\) to represent all \(x\) values satisfying the inequality \((x+2)(x-3)<0\).

Key Concepts

Zero Points

In solving a nonlinear inequality like \((x+2)(x-3) < 0\), identifying the zero points is an essential first step. Zero points are the values of \(x\) where each factor of the inequality equals zero. Solving for these zero points is crucial because they segment the number line into distinct intervals that need to be analyzed separately.

For the factors \(x + 2\) and \(x - 3\), solving \(x + 2 = 0\) gives \(x = -2\), and solving \(x - 3 = 0\) yields \(x = 3\). The number line is therefore divided at these critical points. These divisions help us understand where the inequality might change sign. This is why determining zero points is the foundation of analyzing the inequality.

Interval Notation

Once you've found the zero points, you can segment your number line into intervals for analysis. Interval notation provides a concise way to represent subsets of the real number line.

In our example, the zero points \(-2\) and \(3\) divide the line into three intervals:

• \((-\infty, -2)\)
• \((-2, 3)\)
• \((3, \infty)\)

Each interval describes a range of \(x\) that can be tested to determine where the inequality \((x+2)(x-3)<0\) holds true.By correctly using interval notation, you provide a clear, easy-to-read format that specifies where to focus your test points.

Test Intervals

Testing each interval is key to understanding where the inequality holds true. You select a test point within each interval to determine whether the expression \((x+2)(x-3)\) yields a positive or negative result. The sign of the expression informs you if the inequality condition \(< 0\) is fulfilled in that interval.

For our intervals, choose:

• In \((-\infty, -2)\): Test \(x = -3\), results in a positive value, so the inequality is not satisfied here.
• In \((-2, 3)\): Test \(x = 0\), results in a negative value, which satisfies the inequality.
• In \((3, \infty)\): Test \(x = 4\), results in a positive value, hence not satisfying the inequality.

This method gives you the confidence to establish which intervals make the inequality true.

Graphing Solution Set

Visualizing your answer helps to understand the solution set clearly and effectively. For this problem, after determining that the inequality is satisfied in the interval \((-2, 3)\), mark this on the number line.

To graph the solution, draw a number line. Make use of open circles at \(-2\) and \(3\)since these endpoints are not part of the solution due to the strict "less than" condition. Then, shade the area between these points to represent all the \(x\) values satisfying \((x+2)(x-3)<0\).This simple graphical representation ensures you and others can quickly grasp where the inequality conditions hold and understand the solution visually.