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=blank and x=blank

Answer 1

To solve the equation \(-|11x - 9| + 9 = 7\), we start by isolating the absolute value expression.

1. Subtract 9 from both sides:
\[
-|11x - 9| = 7 - 9
\]
\[
-|11x - 9| = -2
\]

2. Multiply both sides by -1:
\[
|11x - 9| = 2
\]

Now we can split this into two cases based on the definition of absolute value:

**Case 1:**
\[
11x - 9 = 2
\]
Add 9 to both sides:
\[
11x = 11
\]
Divide by 11:
\[
x = 1
\]

**Case 2:**
\[
11x - 9 = -2
\]
Add 9 to both sides:
\[
11x = 7
\]
Divide by 11:
\[
x = \frac{7}{11}
\]

Now we have the two solutions:
- \(x = 1\)
- \(x = \frac{7}{11}\)

To identify the lower and higher-valued solutions:
- The lower-valued solution is \(\frac{7}{11}\)
- The higher-valued solution is \(1\)

So, the final answers are:

Lower-valued solution: \(\frac{7}{11}\)

Higher-valued solution: \(1\)