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: \(1\)