Question

Solve the equation. Check for extraneous solutions. 9|9-8x|=2x+3
simply answer the question

Answer 1

To solve this equation, we'll first solve for the positive value under the absolute value sign and then solve for the negative value.

For the positive value:
9 - 8x = 2x + 3
To isolate the variable, we'll first subtract 2x from both sides and then add 8x to both sides:
9 - 3 = 2x + 8x
6 = 10x
Now divide both sides by 10:
6/10 = x
0.6 = x

For the negative value:
9 - 8x = -(2x + 3)
To eliminate the negative sign, distribute the negative sign to both terms inside the parentheses:
9 - 8x = -2x - 3
Adding 8x to both sides:
9 = 6x - 3
Next, add 3 to both sides:
12 = 6x
Dividing both sides by 6:
12/6 = x
2 = x

Therefore, the solutions to the equation are x = 0.6 and x = 2.

To check for extraneous solutions, we substitute each found value back into the original equation and ensure it satisfies the equation.
For x = 0.6:
9|9 - 8(0.6)| = 2(0.6) + 3
9|9 - 4.8| = 1.2 + 3
9|4.2| = 1.2 + 3
37.8 = 4.2 + 3
37.8 = 7.2
This is not true, so x = 0.6 is an extraneous solution.

For x = 2:
9|9 - 8(2)| = 2(2) + 3
9|9 - 16| = 4 + 3
9|-7| = 4 + 3
9 * 7 = 4 + 3
63 = 7
This is not true, so x = 2 is also an extraneous solution.

Therefore, there are no valid solutions to the equation.