Solve the equation and then check for extraneous solutions.
`4|3x+4|=4x+8`
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To solve the equation, we can divide both sides by 4 to get rid of the absolute value:
|3x + 4| = x + 2
Next, we can consider two cases:
Case 1: 3x + 4 ≥ 0 (the expression inside the absolute value is nonnegative)
In this case, the equation becomes:
3x + 4 = x + 2
Simplifying:
3x - x = 2 - 4
2x = -2
x = -1
However, we need to check if this solution satisfies the original equation:
4|3(-1) + 4| = 4(-1) + 8
4|1| = -4 + 8
4 = 4
Since the equation is true, x = -1 is a valid solution.
Case 2: 3x + 4 < 0 (the expression inside the absolute value is negative)
In this case, we need to change the sign once we remove the absolute value bars. The equation becomes:
-(3x + 4) = x + 2
Simplifying:
-3x - 4 = x + 2
-3x - x = 2 + 4
-4x = 6
x = -1.5
Now, we need to check if this solution satisfies the original equation:
4|3(-1.5) + 4| = 4(-1.5) + 8
4|-0.5| = -6 + 8
4 = 2
The equation is not true, so x = -1.5 is not a valid solution.
Therefore, the only solution to the equation is x = -1.