Solve the equation and then check for extraneous solutions.
`4|3x+4|=4x+8`
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To solve the equation `4|3x+4|=4x+8`, we will split it into two cases:
Case 1: `3x+4 ≥ 0`
In this case, the absolute value of `3x+4` can be expressed as `(3x+4)`, so the equation becomes:
`4(3x+4) = 4x+8`
Expanding the equation:
`12x + 16 = 4x + 8`
Combining like terms:
`12x - 4x = 8 - 16`
`8x = -8`
Dividing both sides by 8:
`x = -1`
Checking for extraneous solutions:
In the original equation, when `x = -1`, we have:
`4|3(-1)+4| = 4(-1)+8`
`4|-3+4| = -4+8`
`4|1| = 4`
`4(1) = 4`
The left side equals the right side, so `x = -1` is a valid solution in this case.
Case 2: `3x+4 < 0`
In this case, the absolute value of `3x+4` can be expressed as `-(3x+4)`, so the equation becomes:
`4-(3x+4) = 4x+8`
Expanding the equation:
`4 - 3x - 4 = 4x + 8`
Combining like terms:
`-3x = 4x + 8 - 4`
`-3x = 4x + 4`
Adding 3x to both sides:
`0 = 7x + 4`
Subtracting 7x from both sides:
`-7x = 4`
Dividing both sides by -7:
`x = -4/7`
Checking for extraneous solutions:
In the original equation, when `x = -4/7`, we have:
`4|3(-4/7)+4| = 4(-4/7)+8`
`4|-12/7+4| = -16/7 + 8`
`4|-12/7 + 4| = -16/7 + 56/7`
`4|(-12 + 28)/7| = 40/7`
`4(16/7) = 40/7`
The left side equals the right side, so `x = -4/7` is also a valid solution in this case.
Therefore, the equation `4|3x+4|=4x+8` has two valid solutions: `x = -1` and `x = -4/7`.