Solve the equation |3x + 4| = 2.

Step-by-step solution

1. 1

   Check that a solution can exist at all. An absolute value is never negative, so $|A| = k$ has solutions only when $k \ge 0$. Here $k = 2 > 0$, so there will be exactly two — one on each side of the vertex.

2. 2

   Split on the definition. $|A| = 2$ means $A$ sits $2$ units from zero, in either direction:

   $3x + 4 = 2 \qquad \text{or} \qquad 3x + 4 = -2$

   Writing both cases before touching either is what keeps the second solution from being lost.

3. 3

   Solve the positive case.

   $3x + 4 = 2 \;\Longrightarrow\; 3x = -2 \;\Longrightarrow\; x = -\frac{2}{3}$

4. 4

   Solve the negative case.

   $3x + 4 = -2 \;\Longrightarrow\; 3x = -6 \;\Longrightarrow\; x = -2$

5. 5

   Verify both and note the symmetry. $\left|3\left(-\tfrac23\right) + 4\right| = |-2 + 4| = 2$ and $|3(-2) + 4| = |-6 + 4| = 2$. Both roots sit symmetrically about the vertex $x = -\tfrac43$, at distance $\tfrac23$ on either side — the graph of $|3x+4|$ is a V with slope $\pm 3$, and $\tfrac{2}{3}$ is exactly the horizontal run needed for a rise of $2$.