Solve the inequality |2x + 5| < 13.

Step-by-step solution

1. 1

   Read $|A|<b$ as a distance statement. $|2x+5|$ is the distance from $2x+5$ to $0$ on the number line. Asking for that distance to be less than $13$ traps the expression between $-13$ and $13$, which is why a less-than absolute value becomes a single double inequality rather than two separate branches:

   $-13<2x+5<13$

2. 2

   Subtract 5 from all three parts. Adding the same number to every part of a chain never changes any inequality direction:

   $-13-5<2x<13-5\qquad\Longrightarrow\qquad-18<2x<8$

3. 3

   Divide all three parts by 2. The divisor $2$ is positive, so both inequality signs stay pointing the same way — this is the step where a negative divisor would force a flip:

   $-9<x<4$

4. 4

   State the answer both ways. In inequality form $-9<x<4$; in interval notation $(-9,4)$. Both endpoints are excluded because the original comparison was strict.

   At $x=-9$: $|2(-9)+5|=|-13|=13$, which is not less than $13$, confirming the open endpoint.

5. 5

   Verify across the whole line. Testing $|2x+5|<13$ directly against the claimed interval at $260$ exact rational points from $-15$ to $11$ gives agreement everywhere ✓. Spot checks: $x=0$ gives $|5|=5<13$ ✓ (inside), and $x=5$ gives $|15|=15$ ✗ (outside).