Let’s reevaluate the inequality \(5 |9 - 5n| - 7 < 38\).
Step 1: Isolate the absolute value
Add \(7\) to both sides:
\[
5 |9 - 5n| < 45
\]
Now, divide both sides by \(5\):
\[
|9 - 5n| < 9
\]
Step 2: Set up the inequality
An absolute value inequality of the form \(|A| < B\) translates to:
\[
-B < A < B
\]
For our case:
\[
-9 < 9 - 5n < 9
\]
Step 3: Solve the two parts of the compound inequality
Part 1:
\[
-9 < 9 - 5n
\]
Subtract \(9\) from both sides:
\[
-18 < -5n
\]
Now, divide by \(-5\) (remember to flip the inequality):
\[
\frac{18}{5} > n \quad \text{(or equivalently)} \quad n < \frac{18}{5}
\]
Part 2:
\[
9 - 5n < 9
\]
Subtract \(9\) from both sides:
\[
-5n < 0
\]
Now divide by \(-5\) (again flipping the inequality):
\[
n > 0
\]
Step 4: Combine the results
So far, we have:
- From Part 1: \(n < \frac{18}{5} \) (or \(n < 3.6\))
- From Part 2: \(n > 0\)
Final Interval Solution
Combining these inequalities gives us:
\[
0 < n < \frac{18}{5}
\]
So the solution is:
\[
(n \in (0, 3.6))
\]
Finding the Other Case
We need to take into account the case for the absolute value when it is negative.
Step 5: Absolute Value Greater Than
In addition to the less than case, we also look at when \( |9 - 5n| \) is greater than \(9\):
\[
|9 - 5n| > 9
\]
This splits into two cases:
-
Case 1:
\[
9 - 5n > 9
\]
\(-5n > 0\) thus \(n < 0\) (This will give us negative values)
-
Case 2:
\[
9 - 5n < -9
\]
Rearranging gives us:
\[
-5n < -18 \implies n > \frac{18}{5}
\]
Final Combined Solutions
Combining both parts:
- From the case \(0 < n < \frac{18}{5}\)
- From the case \(n < 0\) (which is out of the valid \(n\) range)
- And \(n > \frac{18}{5}\), which means \(n > 3.6\)
So, the complete solution set is:
- \( (0, \frac{18}{5}) \)
- \( (3.6, \infty) \)
Thus, the complete solution can be represented as:
\[
\boxed{(0, \frac{18}{5}) \cup (\frac{18}{5}, \infty)}
\]
2 answers