Answer:
Explanation:
To solve the equation 4|3x+4|=4x+8, we need to eliminate the absolute value and solve for x. Let's break it down step-by-step:
Step 1: Remove the absolute value symbols by considering two cases:
Case 1: 3x + 4 ≥ 0
In this case, the absolute value of 3x + 4 remains as is: 4|3x+4| = 4(3x + 4)
Case 2: 3x + 4 < 0
In this case, the absolute value of 3x + 4 becomes its negation: 4|3x+4| = 4(-(3x + 4))
Step 2: Simplify the equations:
Case 1: 4(3x + 4) = 4x + 8
Simplify the equation: 12x + 16 = 4x + 8
Combine like terms: 12x - 4x = 8 - 16
Simplify further: 8x = -8
Divide both sides by 8: x = -1
Case 2: 4(-(3x + 4)) = 4x + 8
Simplify the equation: -12x - 16 = 4x + 8
Combine like terms: -12x - 4x = 8 + 16
Simplify further: -16x = 24
Divide both sides by -16: x = -1.5
Step 3: Check for extraneous solutions by substituting the obtained values back into the original equation.
For x = -1:
4|3(-1)+4| = 4(-1) + 8
4|1| = -4 + 8
4(1) = 4
4 = 4 (True)
For x = -1.5:
4|3(-1.5)+4| = 4(-1.5) + 8
4|0.5| = -6 + 8
4(0.5) = 2
2 = 2 (True)
Since both values satisfy the original equation, there are no extraneous solutions.
Therefore, the solutions to the equation 4|3x+4|=4x+8 are x = -1 and x = -1.5.