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Chapter 11: Problem 6

Solve each inequality, and graph the solution set. $$ (x+6)(x-2)>0 $$

Short Answer

Expert verified
The solution set is \((-fty, -6) \cup (2, fty)\).

Step by step solution

01

Find the roots

Set each factor equal to zero to find the roots: \(x + 6 = 0\) and \(x - 2 = 0\). Solving these, we get: \(x = -6\) and \(x = 2\).
02

Determine the intervals to test

The roots split the number line into three intervals: \((-\infty, -6)\), \((-6, 2)\), \((2, \infty)\). These are the intervals that need to be tested.
03

Test an interval

Choose a test point from each interval and plug it into the inequality \((x+6)(x-2) > 0\): For \(x < -6\), choose \( x = -7\): \((-7+6)(-7-2) = -1(-9) = 9 > 0\), so this interval is part of the solution. For \(-6 < x < 2\), choose \( x = 0\): \((0+6)(0-2) = 6(-2) = -12 < 0\), so this interval is not part of the solution. For \(x > 2\), choose \( x = 3\): \((3+6)(3-2) = 9(1) = 9 > 0\), so this interval is part of the solution.
04

Write the solution set

The solution to the inequality is the union of the intervals where the product is positive: \( (-\infty, -6) \cup (2, \infty) \)
05

Graph the solution set

On a number line, shade the intervals \((-\infty, -6)\) and \((2, \infty)\). Use open circles at \(-6\) and \(2\) to indicate these points are not included in the solution set.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

roots of equations
In solving inequalities, the first key step is to find the roots of the equation. The roots are the values of the variable that make the expression equal to zero. For example, consider the inequality \((x+6)(x-2)>0\). To find the roots, we set each factor equal to zero:
interval testing
Next, we need to determine the intervals to test along the number line. Using the roots \( x = -6 \; \text{and} \; x = 2\), we divide the number line into the following intervals:
graphing solutions
To visualize the solution set of an inequality, it is extremely helpful to graph the intervals on a number line. Once you have tested the intervals, plot the solution set. For our inequality \((x+6)(x-2)>0\), the intervals \((-\frac {6}) \in (\text{is part} \text{inter vol})\frac{2})\)).
This means our solution set is the union of the intervals where the inequality is true:

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