All real numbers correspond to points on the number line, and every point on the number line represents a real number. Real numbers include integers, fractions, decimals, and irrational numbers like √2.

A) The Number Line

Numbers increase from left to right. Points to the left of zero are negative numbers; points to the right of zero are positive numbers. Zero is neither positive nor negative.

• Every real number except zero is either positive or negative.
• From the number line: -4 < -3 < -3/2 < -1, and 1 < √2 < 2.

B) Ordering and "Between"

On the number line, each number is less than any number to its right.

• Strict between: if n is between 1 and 4, then 1 < n < 4 (endpoints excluded).
• Inclusive between: if n is "between 1 and 4, inclusive," then 1 ≤ n ≤ 4 (endpoints included).
• This distinction matters on DS questions — check whether the problem says "inclusive" or not before testing boundary values.

C) Absolute Value

The absolute value of a real number x, written |x|, is defined as:

• |x| = x   if x ≥ 0
• |x| = −x   if x < 0

Geometrically: |x| is the distance between x and zero on the number line. Thus |-3| = |3| = 3 — both are three units from zero. The absolute value of any nonzero number is positive.

Distance interpretation: |x − a| = the distance between x and a on the number line. So |x − 3| < 2 means "x is within 2 units of 3," which gives 1 < x < 5.

Key trap: √(x²) = |x|, not just x. If x = −4, then x² = 16 and √16 = 4 = |−4|.

D) Triangle Inequality

For any real numbers x and y: |x + y| ≤ |x| + |y|.

The absolute value of a sum is never greater than the sum of absolute values. Equality holds when x and y have the same sign (or one is zero).

• If x = 10 and y = 2: |x + y| = |12| = 12 = |x| + |y| ✓ (both positive, so equality holds)
• If x = 10 and y = −2: |x + y| = |8| = 8 < 12 = |x| + |y| ✓ (opposite signs, strict inequality)

E) Absolute Value — 3-Step Method

For equations or inequalities with absolute value expressions, follow three steps:

1. Open the modulus: split into two cases based on the sign of the expression inside.
2. Set conditions: for each case, state the condition (expression ≥ 0 or expression < 0) and solve.
3. Check solutions: verify each solution satisfies the condition assumed in that case. Discard solutions that violate their condition.

Example: |2x − 4| = x + 1.

• Case 1 (2x−4 ≥ 0, i.e., x ≥ 2): 2x−4 = x+1 → x = 5. Check: 5 ≥ 2 ✓
• Case 2 (2x−4 < 0, i.e., x < 2): −(2x−4) = x+1 → 4−2x = x+1 → x = 1. Check: 1 < 2 ✓
• Both solutions valid: x = 5 or x = 1.

F) Absolute Value Equations

|expression| = k means the expression equals k OR equals −k. Always solve both cases.

• |x − 3| = 5 → x − 3 = 5 → x = 8; or x − 3 = −5 → x = −2. Two solutions.
• |expression| = negative number → no solution (absolute value is never negative).
• |x − a| geometrically = distance between x and a on the number line.

G) Absolute Value Inequalities

• |x| < 5 → −5 < x < 5 (bounded single interval — "and" condition)
• |x| > 5 → x < −5 OR x > 5 (two unbounded intervals — "or" condition)
• |x − a| < d → a − d < x < a + d (x is within distance d of a)
• |x − a| > d → x < a − d OR x > a + d (x is more than distance d from a)
• Mnemonic: less than → middle interval; greater than → outer two regions.