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

Question

. 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 can start by isolating the absolute value.

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

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

Next, we need to consider the two cases for the absolute value equation \( |A| = B \) which leads to \( A = B \) or \( A = -B \).

Case 1:

\[ 11x - 9 = 2 \] Adding 9 to both sides: \[ 11x = 2 + 9 \] \[ 11x = 11 \] Dividing by 11: \[ x = 1 \]

Case 2:

\[ 11x - 9 = -2 \] Adding 9 to both sides: \[ 11x = -2 + 9 \] \[ 11x = 7 \] Dividing by 11: \[ x = \frac{7}{11} \]

Now we have two solutions:

1. \( x = 1 \)
2. \( x = \frac{7}{11} \)

Since \( 1 > \frac{7}{11} \), we will write the lower-valued solution first, followed by the higher-valued solution.

Thus, the solutions to the absolute value equation are:

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

In the respective response fields:

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