Certainly! To solve the inequality [tex]\((x - 1)(x + 6) > 0\)[/tex], we need to follow a series of systematic steps. Let's go through them in detail.
### Step 1: Identify the Critical Points
The critical points of the inequality are the roots of the equation [tex]\((x - 1)(x + 6) = 0\)[/tex]. These are the points where the expression changes sign. By setting each factor equal to zero, we find:
1. [tex]\(x - 1 = 0 \implies x = 1\)[/tex]
2. [tex]\(x + 6 = 0 \implies x = -6\)[/tex]
So, the critical points are [tex]\(x = 1\)[/tex] and [tex]\(x = -6\)[/tex].
### Step 2: Divide the Number Line
We will divide the number line into intervals based on these critical points. The intervals to consider are:
1. [tex]\((-\infty, -6)\)[/tex]
2. [tex]\((-6, 1)\)[/tex]
3. [tex]\((1, \infty)\)[/tex]
### Step 3: Test Points in Each Interval
We need to determine the sign of the expression [tex]\((x - 1)(x + 6)\)[/tex] in each interval by choosing a test point from each interval and substituting it into the expression.
- For [tex]\((-\infty, -6)\)[/tex], let's choose [tex]\(x = -7\)[/tex]:
[tex]\[
(-7 - 1)(-7 + 6) = (-8)(-1) = 8 > 0
\][/tex]
The expression is positive in this interval.
- For [tex]\((-6, 1)\)[/tex], let's choose [tex]\(x = 0\)[/tex]:
[tex]\[
(0 - 1)(0 + 6) = (-1)(6) = -6 < 0
\][/tex]
The expression is negative in this interval.
- For [tex]\((1, \infty)\)[/tex], let's choose [tex]\(x = 2\)[/tex]:
[tex]\[
(2 - 1)(2 + 6) = (1)(8) = 8 > 0
\][/tex]
The expression is positive in this interval.
### Step 4: Combine the Intervals
We are interested in the intervals where the expression [tex]\((x - 1)(x + 6)\)[/tex] is greater than zero. From our tests, these intervals are:
- [tex]\((-\infty, -6)\)[/tex]
- [tex]\((1, \infty)\)[/tex]
### Step 5: Create the Number Line Graph
To graph the solution on the number line, we will:
- Use open circles at [tex]\(x = -6\)[/tex] and [tex]\(x = 1\)[/tex] to indicate that these points are not included in the solution (since the inequality is strict: [tex]\(>\)[/tex], not [tex]\(\geq\)[/tex]).
- Shade the regions [tex]\((-\infty, -6)\)[/tex] and [tex]\((1, \infty)\)[/tex].
Here is a visual representation:
```
<=====================(=====)---------------->
-\infty -6 1 \infty
```
Where the parentheses [tex]\(\text{(}\text{)}\)[/tex] indicate the critical points [tex]\(x = -6\)[/tex] and [tex]\(x = 1\)[/tex], which are not included in the solution.
### Final Answer
The solution to the inequality [tex]\((x - 1)(x + 6) > 0\)[/tex] is:
[tex]\[
x \in (-\infty, -6) \cup (1, \infty)
\][/tex]
