Absolute Value Equations

The absolute value of a number is simply its distance from zero, so $|-5| = 5$ and $|5| = 5$. An absolute value equation like $|2x - 3| = 7$ asks: what makes $2x - 3$ land exactly 7 away from zero? The formula up top has three pieces: $E$ (whatever is inside the bars), $b$ (the distance) and "$E = b$ or $E = -b$" (the two spots that far from zero, one on each side). That's all there is to it.

The distance picture. It helps to read $|x - c|$ as "the distance from $x$ to $c$." Then $|x - c| = d$ says $x$ is $d$ units from $c$, and there are exactly two such points, $c - d$ and $c + d$.

Steps. Isolate the absolute value first, then split into two equations.

• If $b < 0$, there's no solution, since distances aren't negative.
• $|A| = |B|$ means $A$ and $B$ are the same distance from zero, so either they're equal or they're opposites: $A = B$ or $A = -B$.

A common mistake. In $3|x + 1| - 2 = 10$, it's tempting to split right away into $3(x + 1) - 2 = \pm10$. That gives the wrong second answer. Isolate the bars first, $|x + 1| = 4$, and only then split.

Manufacturing tolerances ("within 0.01 mm of spec") and error margins in polls are written with absolute values.