Use what you learned in the previous lesson to solve real-world problems.

Check what you understood with a short quiz.

Turn statements like a < x ≤ b into interval notation and a number-line picture. Distinguish open endpoints, closed endpoints, half-open intervals, and rays.

Use what you learned in the previous lesson to solve real-world problems.

Decide whether an interval has finite endpoints on both sides, only one side, or neither side. Use that information to recognize bounded, unbounded, open, closed, and half-open intervals quickly.

Use what you learned in the previous lesson to solve real-world problems.

Check what you understood with a short quiz.

Use |x| as the distance from x to 0, not just as a symbol that makes numbers positive. Translate absolute value into cases: x when x is nonnegative and -x when x is negative.

Use what you learned in the previous lesson to solve real-world problems.

Measure how far apart two real numbers are with |x - y|. Check that distance is always nonnegative, symmetric, and equal to 0 only when the two numbers are the same.

Use what you learned in the previous lesson to solve real-world problems.

Check what you understood with a short quiz.

Translate “x is within r of a” into |x - a| < r and into the interval (a - r, a + r). Read the same idea from a picture, an inequality, or a distance statement.

Use what you learned in the previous lesson to solve real-world problems.

Solve statements like |x - a| > r by identifying the two outside regions beyond a - r and a + r. Contrast being farther than r from a point with being inside a radius around it.

Use what you learned in the previous lesson to solve real-world problems.

Check what you understood with a short quiz.

Treat an epsilon neighborhood of a as all points whose distance from a is less than ε. Connect the notation |x - a| < ε with the open interval centered at a.

Use what you learned in the previous lesson to solve real-world problems.

Recognize 0 < |x - a| < ε as “close to a, but not equal to a.” Use punctured neighborhoods when the point a itself must be excluded while nearby points remain allowed.

Use what you learned in the previous lesson to solve real-world problems.

Check what you understood with a short quiz.

Given an open interval and a point inside it, choose a small radius that keeps nearby points inside the interval. Use the shorter distance to the two endpoints as the safe margin.

Use what you learned in the previous lesson to solve real-world problems.

Use the triangle inequality to control one distance by passing through an intermediate point. Reason from “x is close to y” and “y is close to z” to a bound on how close x is to z.

Use what you learned in the previous lesson to solve real-world problems.

Check what you understood with a short quiz.

Find a new real number between any two different real numbers by taking their midpoint. Use this to see why the real line has no “next” point after a given number.

Use what you learned in the previous lesson to solve real-world problems.

Check what you understood with a short quiz.

Review this chapter with practice based on your mistakes.