Solve the equation. Check for extraneous solutions.

Question

9|9 - 8x| = 2x + 3

Answer 1

Let's solve the equation step by step:

1. Remove the absolute value by splitting it into two separate cases:
9 - 8x = 2x + 3 or 9 - 8x = -(2x + 3)

2. Solve the first case:
9 - 8x = 2x + 3

Move all terms with x to one side:
9 - 3 = 2x + 8x

Combine like terms:
6 = 10x

Divide by 10:
6/10 = x
Simplify: 3/5 = x

3. Solve the second case:
9 - 8x = -(2x + 3)

Remove the negative sign by multiplying both sides by -1:
-(9 - 8x) = -(-2x - 3)

Distribute the negative sign:
-9 + 8x = 2x + 3

Move all terms with x to one side:
-9 - 3 = 2x - 8x

Combine like terms:
-12 = -6x

Divide by -6:
-12/-6 = x
Simplify: 2 = x

Checking for extraneous solutions:
Plug x = 3/5 into the original equation:
9|9 - 8(3/5)| = 2(3/5) + 3

Simplify:
9|9 - 24/5| = 6/5 + 3

Evaluate the absolute value:
9|45/5 - 24/5| = 6/5 + 3

Simplify:
9|21/5| = 6/5 + 15/5

Evaluate the absolute value:
9(21/5) = 21/5 + 15/5

Simplify:
189/5 = 36/5

Since the equation is not true, x = 3/5 is not a valid solution.

Plug x = 2 into the original equation:
9|9 - 8(2)| = 2(2) + 3

Simplify:
9|9 - 16| = 4 + 3

Evaluate the absolute value:
9|-7| = 4 + 3

Simplify:
9(7) = 7

Since the equation is true, x = 2 is a valid solution.

Therefore, the solution to the equation is x = 2.