Absolute Value Inequalities

An absolute value inequality is simply a question about distance: "which numbers are within 3 of 4?" or "which are more than 5 away from 0?" Machinists live by these; a part that must be "50 mm, give or take 0.2" is $|L - 50| \le 0.2$. The formulas up top have the same small pieces as last lesson, $E$ (what's inside the bars) and $b$ (the distance), and they just translate "closer than" and "farther than" into ordinary inequalities. Picture the number line and it's almost obvious.

"Less than" makes an AND. Being within $b$ of zero means being above $-b$ and below $b$ at the same time. That's one interval, written as a double inequality, like the ones from the linear inequalities lesson.

"Greater than" makes an OR. Being more than $b$ from zero means you're either far out to the left or far out to the right, never both. That's two separate rays.

Edge cases. $|E| < b$ with $b \le 0$ has no solutions, since no distance is less than zero (or less than a negative number). And $|E| > b$ with $b < 0$ is true for every $x$, since every distance is at least zero.

A common mistake. It's tempting to write $|2x + 1| > 5$ as $-5 > 2x + 1 > 5$, by copying the "less than" pattern. That says a number is below $-5$ and above $5$ at once, which nothing can be. "Greater than" has to split into two separate inequalities joined by "or."

Quality control (parts within tolerance), signal thresholds and confidence intervals in statistics are absolute value inequalities.