Solve the absolute value equation −|11x−9|+9=7

Question

−
|
11
𝑥
−
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=
𝑥
=
and x=
𝑥
=
.

Answer 1

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

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

Subtract 9 from both sides:

\[ -|11x-9| = 7 - 9 \] \[ -|11x-9| = -2 \]

Now multiply both sides by -1:

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

Next, we can remove the absolute value by considering the two cases:

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

Solving the first case:

\[ 11x - 9 = 2 \] Add 9 to both sides:

\[ 11x = 11 \] Divide by 11:

\[ x = 1 \]

Solving the second case:

\[ 11x - 9 = -2 \] 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 provide the answers in the required format, we identify the lower-valued solution and the higher-valued solution:

• Lower-valued solution: \(\frac{7}{11}\)
• Higher-valued solution: \(1\)

So, the final answers are:

• \(x = \frac{7}{11}\) (lower-valued solution)
• \(x = 1\) (higher-valued solution)