6.3 Absolute Value, Number Line, Decimals & Sequences

Absolute Value, Number Line, Decimals & Sequences

Quick Answer: Absolute value measures distance from zero (always nonnegative), the number line runs rightward positive, decimal representation splits into terminating and repeating forms, and sequences come in arithmetic (common difference d), geometric (common ratio r), and special (Fibonacci) families. GRE QC questions on these topics hinge on whether a value can be negative, zero, or positive.

Absolute Value

|x| = x if x ≥ 0, and |x| = −x if x < 0. Three properties you must internalize:

• |x| ≥ 0 for every real x.
• |x| = |−x| (symmetry).
• Triangle inequality: |a + b| ≤ |a| + |b|.

Solving |x − 3| = 5 means x is 5 units from 3: x = 8 or x = −2. Solving |x| < 4 means −4 < x < 4 (an interval, not a single value).

Worked Example — Absolute Value Inequality

Solve |2x − 5| ≤ 7.

• Rewrite as −7 ≤ 2x − 5 ≤ 7.
• Add 5: −2 ≤ 2x ≤ 12.
• Divide by 2: −1 ≤ x ≤ 6.
• Solution: every x in [−1, 6].

For QC, remember |x| can be 0 — so "Quantity A = |a − b|" could be 0 when a = b. Never assume |expression| is strictly positive; if Quantity B is 0, test whether a = b is possible.

Worked Example — QC Trap with Absolute Value

Quantity A: |a − b|. Quantity B: 0. If a and b are real numbers with no other constraint, can |a − b| equal 0? Yes, when a = b. Can it exceed 0? Yes, when a ≠ b. So sometimes A = B (both 0) and sometimes A > B; the answer is (D) — the relationship cannot be determined.

The Number Line

The positive direction is to the right (ETS convention). Integers are evenly spaced. Fractions and decimals occupy positions between integers. The distance between a and b is |b − a|, always nonnegative.

Worked Example — Distance and Midpoint

On a number line, points at −3 and 7. Distance = |7 − (−3)| = 10. Midpoint = (−3 + 7)/2 = 2.

GRE often plots variables a, b, c on a line and asks comparisons. If the figure shows b between a and c, then a < b < c. But geometric figures are NOT necessarily drawn to scale for lengths — only the order is reliable unless explicitly stated. Coordinate axes and bar/circle/line graphs, by contrast, ARE drawn to scale.

Decimal Representation

Every rational number has a decimal expansion that is either terminating (1/4 = 0.25) or repeating (1/3 = 0.333..., 1/7 = 0.142857142857...).

A fraction p/q in lowest terms terminates if and only if q has no prime factors other than 2 and 5. So 1/8 (q = 2³) terminates, but 1/6 (q = 2 × 3) repeats.

Worked Example — Convert 0.72̄ (0.7222...) to a Fraction

• Let x = 0.7222...
• 10x = 7.222...
• 100x = 72.222...
• Subtract: 100x − 10x = 72.222... − 7.222... = 65 → 90x = 65 → x = 65/90 = 13/18.

For a purely repeating decimal with period k, the denominator is (10^k − 1): 0.3636... = 36/99 = 4/11. For a mixed decimal like 0.72̄, multiply by 10^(number of non-repeating digits) and again by 10^(number of repeating digits), then subtract.

Sequences

Three families dominate GRE sequence questions.

Arithmetic Sequences

Each term differs from the previous by a constant d. nth term: a_n = a_1 + (n − 1)d. Sum of first n terms: S_n = n/2 × (a_1 + a_n).

Worked example: 5, 9, 13, 17, ... Find the 20th term.

• a_1 = 5, d = 4. a_20 = 5 + 19 × 4 = 5 + 76 = 81.
• Sum of first 20: S_20 = 20/2 × (5 + 81) = 10 × 86 = 860.

Geometric Sequences

Each term is r times the previous. nth term: a_n = a_1 × r^(n−1). Sum of first n terms: S_n = a_1 × (r^n − 1) / (r − 1), for r ≠ 1.

Worked example: 3, 6, 12, 24, ... Find a_6.

• a_1 = 3, r = 2. a_6 = 3 × 2⁵ = 3 × 32 = 96.

Special Sequences

The Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, ... each term is the sum of the two preceding. The GRE usually defines any nonstandard sequence explicitly ("a_n = a_(n−1) + a_(n−2) − 1"); always plug the given recurrence rather than assume a familiar pattern.

Worked Example — Recurrence Plug-In

A sequence is defined by a_1 = 2 and a_n = 3a_(n−1) − 1. Find a_4.

• a_2 = 3(2) − 1 = 5.
• a_3 = 3(5) − 1 = 14.
• a_4 = 3(14) − 1 = 41.

GRE sequences reward careful indexing — write a_1, a_2, a_3 explicitly before trusting a closed-form formula. A common error is to use a_n = a_1 + (n − 1)d for a recurrence that is NOT arithmetic; if the question defines the sequence recursively, compute term by term instead.