Solve the absolute value equation |2x + 5| = 0.

Step-by-step solution

1. 1

   Notice that the usual two-case split degenerates here. For $|A|=c$ with $c>0$ you write $A=c$ or $A=-c$. With $c=0$ those two cases are the same equation, because $+0$ and $-0$ are the same number.

2. 2

   Drop the bars. The only real number at distance $0$ from the origin is $0$ itself, so:

   $|2x+5|=0\iff 2x+5=0$

3. 3

   Solve the resulting linear equation. Subtract $5$ and divide by $2$:

   $2x=-5\quad\Longrightarrow\quad x=-\frac{5}{2}=-2.5$

4. 4

   Check and interpret. Substituting back gives $|2(-2.5)+5|=|0|=0$ $\checkmark$. Geometrically, the graph of $y=|2x+5|$ is a V whose vertex sits on the $x$-axis at $x=-\tfrac52$; the equation asks where that V touches height $0$, and a V touches its minimum at exactly one point.