To solve the inequality [tex]\((x+7)(x+5)(x-7) < 0\)[/tex] and graph the solution set, follow these steps:
### Step-by-Step Solution:
1. Identify the Critical Points:
The critical points are the values of [tex]\( x \)[/tex] that make each factor equal to zero. These points divide the number line into intervals where the expression can change its sign.
For the factors [tex]\( x + 7 \)[/tex], [tex]\( x + 5 \)[/tex], and [tex]\( x - 7 \)[/tex]:
[tex]\[
x + 7 = 0 \implies x = -7
\][/tex]
[tex]\[
x + 5 = 0 \implies x = -5
\][/tex]
[tex]\[
x - 7 = 0 \implies x = 7
\][/tex]
Thus, the critical points are [tex]\( x = -7 \)[/tex], [tex]\( x = -5 \)[/tex], and [tex]\( x = 7 \)[/tex].
2. Determine the Intervals:
The critical points divide the number line into four intervals:
[tex]\[
(-\infty, -7), \quad (-7, -5), \quad (-5, 7), \quad (7, \infty)
\][/tex]
3. Test Each Interval:
Choose a test point from each interval and substitute it into the inequality to determine if the product is positive or negative.
- Interval [tex]\( (-\infty, -7) \)[/tex]: Choose [tex]\( x = -8 \)[/tex]
[tex]\[
(x+7)(x+5)(x-7) = (-8+7)(-8+5)(-8-7) = (-1)(-3)(-15) = -45 \quad (\text{negative})
\][/tex]
- Interval [tex]\( (-7, -5) \)[/tex]: Choose [tex]\( x = -6 \)[/tex]
[tex]\[
(x+7)(x+5)(x-7) = (-6+7)(-6+5)(-6-7) = (1)(-1)(-13) = 13 \quad (\text{positive})
\][/tex]
- Interval [tex]\( (-5, 7) \)[/tex]: Choose [tex]\( x = 0 \)[/tex]
[tex]\[
(x+7)(x+5)(x-7) = (0+7)(0+5)(0-7) = (7)(5)(-7) = -245 \quad (\text{negative})
\][/tex]
- Interval [tex]\( (7, \infty) \)[/tex]: Choose [tex]\( x = 8 \)[/tex]
[tex]\[
(x+7)(x+5)(x-7) = (8+7)(8+5)(8-7) = (15)(13)(1) = 195 \quad (\text{positive})
\][/tex]
4. Determine Where the Inequality is Satisfied:
We need the intervals where the product is less than zero (negative):
[tex]\[
(-\infty, -7) \quad \text{and} \quad (-5, 7)
\][/tex]
5. Combine the Intervals:
The combined solution in interval notation is:
[tex]\[
(-\infty, -7) \cup (-5, 7)
\][/tex]
### Graphical Representation:
To visualize the solution set, we plot it on the number line.
1. Draw a number line.
2. Mark the critical points [tex]\( x = -7 \)[/tex], [tex]\( x = -5 \)[/tex], and [tex]\( x = 7 \)[/tex].
3. Shade the intervals [tex]\((-∞, -7)\)[/tex] and [tex]\((-5, 7)\)[/tex]. Use open circles at the points [tex]\( x = -7, -5, 7 \)[/tex] to indicate that these points are not included in the solution set.
Here’s how the graph looks:
```
----o========o------------------o===========o----
-∞ -7 -5 7 ∞
```
In summary, the solution to the inequality [tex]\((x+7)(x+5)(x-7) < 0\)[/tex] is:
[tex]\[
x \in (-\infty, -7) \cup (-5, 7)
\][/tex]
