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Solve the equation. Check for extraneous solutions.
9|9−8x|=2x+3
To solve the equation 9|9-8x| = 2x + 3:
1. Let's start by isolating the absolute value term. We can do this by dividing both sides by 9:
|9-8x| = (2x + 3) / 9
2. There are two possibilities for the absolute value:
a. 9-8x = (2x + 3) / 9
b. 9-8x = -((2x + 3) / 9)
3. Solve for x in each case:
a. 9-8x = (2x + 3) / 9
9 - 8x = 2x/9 + 1/3
27 - 24x = 2x + 3
27 = 26x + 3
24 = 26x
x = 24 / 26
x = 12 / 13
b. 9-8x = -((2x + 3) / 9)
9-8x = -(2x / 9) - 1/3
27 - 24x = -2x - 9
27 = 22x - 9
36 = 22x
x = 36 / 22
x = 18 / 11
4. Check for extraneous solutions by plugging the values of x back into the original equation:
For x = 12/13:
9|9 - 8(12/13)| = 2(12/13) + 3
9|9 - 96/13| = 24/13 + 3
9|117/13| = 24/13 + 3
9(117/13) = 24/13 + 39/13
1053/13 = 63/13
True
For x = 18/11:
9|9 - 8(18/11)| = 2(18/11) + 3
9|9 - 144/11| = 36/11 + 3
9|99/11 - 144/11| = 36/11 + 3
9|-45/11| = 36/11 + 3
9(45/11) = 36/11 + 33/11
405/11 = 69/11
False
Therefore, the solution to the equation is x = 12/13.