Question

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Part 1
Solve the polynomial inequality and graph the solution set on a real number line. Express the solution set in interval notation.
left parenthesis x minus 6 right parenthesis left parenthesis x plus 9 right parenthesis greater than 0
Question content area bottom
Part 1
Solve the inequality. What is the solution​ set? Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The solution is
  
enter your response here.
​(Simplify your answer. Type your answer in interval notation. Type an exact​ answer, using radicals as needed. Use integers or fractions for any numbers in the​ expression.)
B.
The solution set is the empty set.

Answer 1

To solve the polynomial inequality \((x - 6)(x + 9) > 0\), we will first identify the roots of the equation by setting the left-hand side equal to zero:

\[
(x - 6)(x + 9) = 0
\]

This gives us two roots:

1. \(x - 6 = 0 \implies x = 6\)
2. \(x + 9 = 0 \implies x = -9\)

Next, we will use these roots to determine the intervals where the product \((x - 6)(x + 9)\) is greater than zero. The critical points divide the real number line into three intervals:

1. \((- \infty, -9)\)
2. \((-9, 6)\)
3. \((6, \infty)\)

To find the sign of \((x - 6)(x + 9)\) in each interval, we can test a point from each interval:

1. **Interval \((- \infty, -9)\)**: Choose \(x = -10\)
\[
(-10 - 6)(-10 + 9) = (-16)(-1) = 16 > 0
\]

2. **Interval \((-9, 6)\)**: Choose \(x = 0\)
\[
(0 - 6)(0 + 9) = (-6)(9) = -54 < 0
\]

3. **Interval \((6, \infty)\)**: Choose \(x = 7\)
\[
(7 - 6)(7 + 9) = (1)(16) = 16 > 0
\]

Now we have the signs of \((x - 6)(x + 9)\):
- Positive on \((- \infty, -9)\)
- Negative on \((-9, 6)\)
- Positive on \((6, \infty)\)

We need the intervals where the product is greater than zero:

\[
(-\infty, -9) \cup (6, \infty)
\]

Thus, the solution set in interval notation is:

\[
\boxed{(-\infty, -9) \cup (6, \infty)}
\]