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

Step-by-step solution

1. 1

   Confirm two solutions are expected. The right-hand side $2$ is positive, so the expression inside the bars can be either $+2$ or $-2$. A positive right-hand side always yields two solutions; only a right-hand side of $0$ gives one, and a negative one gives none.

2. 2

   Write the two cases. From $|A|=2$ we get $A=2$ or $A=-2$, with $A=4x+3$:

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

3. 3

   Solve the first case. Subtract $3$, then divide by $4$:

   $4x=-1\quad\Longrightarrow\quad x=-\frac{1}{4}$

4. 4

   Solve the second case. Subtract $3$ from $-2$ this time:

   $4x=-5\quad\Longrightarrow\quad x=-\frac{5}{4}$

5. 5

   Verify both roots. Substituting each value back reproduces $2$:

   $\left|4\left(-\tfrac14\right)+3\right|=|-1+3|=2\qquad\checkmark$
   $\left|4\left(-\tfrac54\right)+3\right|=|-5+3|=|-2|=2\qquad\checkmark$

   Notice the two roots sit symmetrically about $x=-\tfrac34$, the value that makes the inside zero — that symmetry is a useful sanity check for every equation of this shape.