Question

Solve the absolute value equation −|11x−9|+9=7 . Enter the lower-valued solution in the first response field and the higher-valued solution in the second response field. Provide exact solutions in fractional form, where appropriate.(1 point) The solutions to the absolute value equation are x=() and x=()

Answer 1

To solve the equation \(-|11x - 9| + 9 = 7\), we first isolate the absolute value expression:

\[
-|11x - 9| + 9 = 7
\]

Subtract 9 from both sides:

\[
-|11x - 9| = 7 - 9
\]

\[
-|11x - 9| = -2
\]

Multiplying both sides by -1 (this reverses the inequality):

\[
|11x - 9| = 2
\]

Now we will break this absolute value equation into two cases:

1. **Case 1:**
\[
11x - 9 = 2
\]
Solving for \(x\):
\[
11x = 11
\]
\[
x = 1
\]

2. **Case 2:**
\[
11x - 9 = -2
\]
Solving for \(x\):
\[
11x = 7
\]
\[
x = \frac{7}{11}
\]

Thus, the solutions to the absolute value equation are \(x = 1\) and \(x = \frac{7}{11}\).

To provide answers in ascending order:
- The lower-valued solution is \(\frac{7}{11}\).
- The higher-valued solution is \(1\).

So, we have:
- Lower-valued solution: \(\frac{7}{11}\)
- Higher-valued solution: \(1\)