We are solving the equation:

$4|3x + 4| = 4x + 8$

Step-by-Step Solution:

1. Isolate the Absolute Value: Divide both sides of the equation by 4: $|3x + 4| = x + 2$

2. Break into Two Cases: Since the absolute value expression $|3x + 4|$ can equal $x + 2$ or $-(x + 2)$, we write:

   • Case 1: $3x + 4 = x + 2$
   • Case 2: $3x + 4 = -(x + 2)$

3. Solve Each Case:

   Case 1: $3x + 4 = x + 2$ Subtract $x$ from both sides: $2x + 4 = 2$ Subtract 4 from both sides: $2x = -2$ Divide by 2: $x = -1$

   Case 2: $3x + 4 = -(x + 2)$ Distribute the negative sign: $3x + 4 = -x - 2$ Add $x$ to both sides: $4x + 4 = -2$ Subtract 4 from both sides: $4x = -6$ Divide by 4: $x = -\frac{3}{2}$

4. Check for Validity: Substitute $x = -1$ and $x = -\frac{3}{2}$ into the original equation to ensure both solutions satisfy the equation:

   • For $x = -1$: $4|3(-1) + 4| = 4(-1) + 8$ $4| -3 + 4 | = -4 + 8$ $4(1) = 4 \quad \text{(True)}$

   • For $x = -\frac{3}{2}$: $4|3(-\frac{3}{2}) + 4| = 4(-\frac{3}{2}) + 8$ $4| -\frac{9}{2} + 4| = -6 + 8$ $4|-\frac{1}{2}| = 2$ $4(\frac{1}{2}) = 2 \quad \text{(True)}$

Both solutions $x = -1$ and $x = -\frac{3}{2}$ are valid.

Final Answer:

$x = -1 \quad \text{or} \quad x = -\frac{3}{2}$

Would you like additional explanation or visualizations?

Related Questions:

1. How can you graph this equation to verify the solutions?
2. What are some common properties of absolute value equations?
3. Can you explain the cases where absolute values have no solutions?
4. How does the distributive property work when expanding $-(x+2)$?
5. How would this problem change if the equation included $|x - 2|$ instead?

Tip:

Always check your solutions when solving absolute value equations, as extraneous solutions may arise!