Absolute value: Rearranging Equations

Given the equation $2|x + 1| = 8$, divide both sides by 2: $|x + 1| = 4$.

Consider the two cases for the absolute value:

1. Case 1: $x + 1 = 4$
   Subtract 1: $x = 3$.
2. Case 2: $x + 1 = -4$
   Subtract 1: $x = -5$.

Thus, $x$ is $3 \text{ or } -5$.

Given the equation $|z - 7| = 3$, consider the two cases for the absolute value:

1. Case 1: $z - 7 = 3$
   Add 7: $z = 10$.
2. Case 2: $z - 7 = -3$
   Add 7: $z = 4$.

Thus, $z$ is $10 \text{ or } 4$.

To solve the absolute value equation $\left|3x + 1\right| = 4$, we set up two separate equations because absolute value represents the distance from zero, meaning the expression inside can be equal to 4 or -4.

1. First equation: $3x + 1 = 4$
2. Subtract 1 from both sides: $3x = 3$
3. Divide both sides by 3: $x = 1$

2. Second equation: $3x + 1 = -4$
3. Subtract 1 from both sides: $3x = -5$
4. Divide both sides by 3: $x = -\frac{5}{3}$

Therefore, the solutions are $x = 1$ and $x = -\frac{5}{3}$, but only the negative solution satisfies the original setup with a valid subtraction and division sequence again as per the equation.