Modulus inequalities: between, or outside

Modulus inequalities come in two shapes, and they behave differently.

Shape 1: the modulus is alone on the left.

$|x - a| < b$

Read it as "the distance from $x$ to $a$ is less than $b$". This works because $x - a$ is how far $x$ is from $a$, with a sign; the modulus removes the sign. The answer is one interval.

Shape 2: ordinary algebra is on one side.

$2x + 5 < |x + 1|$

Here the modulus is not alone, so you cannot use the interval formula. Split the number line into cases.

Learn the geometry first. An inequality describes a region of the number line, not a single point.

Being within $b$ of $a$ means being between the two points $b$ to the left and $b$ to the right of $a$:

$a - b < x < a + b$

So the method is:

1. Check $b > 0$. If the question says $|x - a| < -2$, the answer is "no solutions".
2. Write the double inequality $a - b < x < a + b$.
3. Simplify the two ends if the arithmetic is not already done.
4. Write the answer as an inequality, not as a list of numbers.

For the reverse direction the answer is two separate pieces instead of one:

$|x - a| > b \iff x < a - b \quad\text{or}\quad x > a + b$

The forgotten step: the direction of the inequality decides the shape of the answer. "Less than" gives the middle piece; "greater than" gives the two outer pieces.