Problem 23 - The Calculus with Analytic Geometry

Question

In Exercises 11 through 32 , find the solution set of the given inequality and illustrate the solution on the real number $$ (x-3)(x+5)>0 $$

Step-by-Step Solution

Answer

The solution set is \( (-\infty, -5) \cup (3, \infty) \).

Step 1: Identify the Zero Points

Determine the values of x that make each factor zero. For \((x-3) = 0\), x = 3. For \( (x+5) = 0 \), x = -5. The zero points are x = 3 and x = -5.

Step 2: Determine the Intervals

The zero points divide the number line into three intervals: \((-\infty, -5) \), \((-5, 3)\), and \( (3, \infty) \).

Step 3: Test the Intervals

Select a test point from each interval to determine if the product \( (x-3)(x+5) \) is positive or negative in that interval.For \( x < -5 \), choose \( x = -6 \): \( (-6-3)(-6+5) = (-9)(-1) = 9 > 0 \).For \( -5 < x < 3 \), choose \( x = 0 \): \( (0-3)(0+5) = (-3)(5) = -15 < 0 \).For \( x > 3 \), choose \( x = 4 \): \( (4-3)(4+5) = (1)(9) = 9 > 0 \).

Step 4: Combine the Solution Intervals

The inequality holds in the intervals where the product is positive. Therefore, the solution set is \( (-\infty, -5) \cup (3, \infty) \).

Step 5: Illustrate the Solution

On a number line, shade the intervals \( (-\infty, -5) \) and \( (3, \infty) \), and use open circles at -5 and 3 to indicate that these points are not included in the solution.

Key Concepts

Inequality Solution Set

When solving any inequality, finding the solution set boils down to determining the range of values that satisfy the given inequality. For our example \((x-3)(x+5)>0\), we need to find the values of x that make the expression positive. This involves identifying the 'zero points' where each factor in the inequality equals zero. In this case, the factors are \((x-3)\) and \((x+5)\). By setting each equal to zero, we find x equals 3 and -5, respectively.

The steps to solve are:

• Identify the zero points, i.e., where each factor equals zero.
• Determine intervals on the number line created by these zero points.
• Test each interval to determine which satisfy the inequality.
• Combine the intervals to form the final solution set.

For the inequality \((x-3)(x+5)>0\), the solution set is \((-\infty, -5) \cup (3, \infty)\). This means x can be any value within these intervals, except for -5 and 3 themselves.

Number Line Illustration

Visualizing the solution set on a number line is an essential step. This helps to clearly see the intervals where the inequality holds true. To represent the solution set \((-\infty, -5) \cup (3, \infty)\):

• Start by marking the zero points: -5 and 3, with open circles to indicate that these points are not included in the solution.
• Shade the interval on the number line from -∞ to -5 (without including -5).
• Leave out the interval between -5 and 3 unshaded because the inequality does not hold true in this range.
• Finally, shade the interval from 3 to ∞ (excluding 3).

This visual representation makes it easier to understand which parts of the number line satisfy the inequality. It helps confirm that the solution set is absolute and comprehensive.

Test Interval Method

The test interval method is a critical technique for determining which intervals formed by zero points satisfy the inequality. Here’s how you use this method:

• First, determine the zero points of the inequality by solving \((x-3) = 0\) and \((x+5) = 0\). These values are x = 3 and x = -5.
• Next, divide the number line into intervals using these zero points: \((-\infty, -5)\), \((-5, 3)\), and \((3, \infty)\).
• Choose a test point from within each interval, such as -6 for \((-\infty, -5)\), 0 for \((-5, 3)\), and 4 for \((3, \infty)\).
• Plug each test point into the inequality \((x-3)(x+5)\) to determine if the product is positive or negative in the interval.

For example, at x = -6, the product \((x-3)(x+5) = ( -6 - 3)(-6 + 5) = (-9)( -1) = 9>0\). Hence, \(( -\infty, -5)\) satisfies the inequality. Likewise, test intervals \((-5, 3)\) and \((3, \infty)\) to find where the inequality holds true.

By testing each interval, you can determine exactly where the inequality is satisfied, contributing to a comprehensive and accurate solution set.