Solve the equation. Check for extraneous solutions.
6|6 – 4x| = 8x + 4
To solve the equation, we need to eliminate the absolute value notation.
We start by isolating the absolute value expression:
|6 - 4x| = (8x + 4) / 6
Next, we consider two cases:
1. 6 - 4x ≥ 0 (positive value inside the absolute value)
In this case, the absolute value expression remains unchanged:
6 - 4x = (8x + 4) / 6
2. 6 - 4x < 0 (negative value inside the absolute value)
In this case, the absolute value expression becomes the negative of its original value:
-(6 - 4x) = (8x + 4) / 6
Case 1: 6 - 4x = (8x + 4) / 6
Multiply both sides by 6 to eliminate the fraction:
36 - 24x = 8x + 4
Combine like terms:
-24x - 8x = 4 - 36
-32x = -32
Divide both sides by -32:
x = 1
Case 2: -(6 - 4x) = (8x + 4) / 6
Expand the parentheses and multiply both sides by 6 to eliminate the fraction:
-6 + 4x = - (8x + 4)
-6 + 4x = -8x - 4
Combine like terms:
4x + 8x = -4 + 6
12x = 2
Divide both sides by 12:
x = 2/12
x = 1/6
Therefore, the solutions to the equation are x = 1 and x = 1/6.
Now we need to check for extraneous solutions by substituting these values back into the original equation:
For x = 1:
6|6 - 4(1)| = 8(1) + 4
6|6 - 4| = 8 + 4
6|2| = 12
12 = 12 (True)
For x = 1/6:
6|6 - 4(1/6)| = 8(1/6) + 4
6|6 - 2/3| = 4/3 + 4
6|18/3 - 2/3| = 16/3
6|16/3| = 16/3
For the absolute value to equal 16/3, the expression inside the absolute value must be either 16/3 or -16/3.
For 6 - 4x = 16/3:
6 - 4(1/6) = 16/3
6 - 2/3 = 16/3
(18 - 2)/3 = 16/3
16/3 = 16/3 (True)
For 6 - 4x = -16/3:
6 - 4(1/6) = -16/3
6 - 2/3 = -16/3
(18 - 2)/3 = -16/3
16/3 = -16/3 (False)
Hence, x = 1/6 is the only solution that is not extraneous.