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 absolute value equation \(-|11x - 9| + 9 = 7\), we first isolate the absolute value term.

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

2. Multiply both sides by -1 (which flips the inequality):
\[
|11x - 9| = 2
\]

Now we can set up two equations based on the definition of absolute value:

1. \(11x - 9 = 2\)
2. \(11x - 9 = -2\)

Let's solve each equation separately.

**For the first equation:**
\[
11x - 9 = 2
\]
Add 9 to both sides:
\[
11x = 11
\]
Divide by 11:
\[
x = 1
\]

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

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

To enter the solutions in the specified format:
- The lower-valued solution \(x = \frac{7}{11}\)
- The higher-valued solution \(x = 1\)

So, the answers are:
1. \(\frac{7}{11}\)
2. \(1\)