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=()

Question

GPT-4o mini · Answer

To solve the equation $-|11x - 9| + 9 = 7$, we first isolate the absolute value expression: $$ -|11x - 9| + 9 = 7 $$ Subtract 9 from both sides: $$ -|11x - 9| = 7 - 9 $$ $$ -|11x - 9| = -2 $$ Multiplying both sides by -1 (this reverses the inequality): $$ |11x - 9| = 2 $$ Now we will break this absolute value equation into two cases: 1. **Case 1:** $$ 11x - 9 = 2 $$ Solving for $x$: $$ 11x = 11 $$ $$ x = 1 $$ 2. **Case 2:** $$ 11x - 9 = -2 $$ Solving for $x$: $$ 11x = 7 $$ $$ x = \frac{7}{11} $$ Thus, the solutions to the absolute value equation are $x = 1$ and $x = \frac{7}{11}$. To provide answers in ascending order: - The lower-valued solution is $\frac{7}{11}$. - The higher-valued solution is $1$. So, we have: - Lower-valued solution: $\frac{7}{11}$ - Higher-valued solution: $1$