2.4 Number Lines, Order of Operations & Estimation

2.4 Number Lines, Order of Operations & Estimation

Precision and efficiency are essential for high GRE Quantitative scores. This section covers order of operations rules, number line geometry, inequality properties, and estimation shortcuts for quickly evaluating complex expressions without unnecessary calculation.

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Order of Operations (PEMDAS)

When evaluating mathematical expressions, operations must be performed in a strict hierarchy governed by PEMDAS:

1. P - Parentheses & Grouping Symbols: Evaluate expressions inside parentheses (), brackets [], braces {}, fraction bars, and radicals from the innermost grouping outward.
2. E - Exponents & Roots: Evaluate all power terms ($x^n$) and radical roots ($\sqrt{x}$).
3. MD - Multiplication & Division: Perform multiplication and division left to right in the exact order they appear. Multiplication does NOT take precedence over division!
4. AS - Addition & Subtraction: Perform addition and subtraction left to right in the exact order they appear.

GRE Order-of-Operations Trap:
Consider $12 \div 3 \times 2$. Because division and multiplication share equal priority left-to-right:
Correct: $(12 \div 3) \times 2 = 4 \times 2 = 8$.
Incorrect: $12 \div (3 \times 2) = 12 \div 6 = 2$.

Worked Example: Complex Operation Evaluation

Evaluate: $18 - 6 \div 2 \times 3 + (4^2 - 10)$

1. Inner Parentheses: $(4^2 - 10) = (16 - 10) = 6$
2. Expression becomes: $18 - 6 \div 2 \times 3 + 6$
3. Multiplication/Division (Left to Right):
   • First: $6 \div 2 = 3$
   • Next: $3 \times 3 = 9$
4. Expression becomes: $18 - 9 + 6$
5. Addition/Subtraction (Left to Right):
   • First: $18 - 9 = 9$
   • Next: $9 + 6 = 15$

• Final Answer: $15$

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Real Number Line Properties & Distance

The real number line maps every real number to a unique point in space. Numbers increase moving to the right and decrease moving to the left.

  -4   -3   -2   -1    0    1    2    3    4
<---|----|----|----|----|----|----|----|--->

Distance & Midpoint Formulas on a Line

For any two points $a$ and $b$ on the real number line:

• Distance Formula: $d(a, b) = |a - b| = |b - a|$
• Midpoint Formula: $M = \frac{a + b}{2}$

Example: Distance between $-5$ and $7$

$d(-5, 7) = | -5 - 7 | = | -12 | = 12$ $\text{Midpoint} = \frac{-5 + 7}{2} = \frac{2}{2} = 1$

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Inequalities on the Number Line

Inequalities describe relative positions on the number line:

• $a < b$: $a$ lies to the left of $b$.
• $a > b$: $a$ lies to the right of $b$.

Key Rules for Inequality Manipulations

1. Addition & Subtraction: Adding or subtracting any real number $c$ preserves the inequality sign:
   $a < b \implies a + c < b + c$
2. Positive Multiplication/Division: Multiplying or dividing by a positive number $c > 0$ preserves the inequality sign:
   $a < b \implies a \cdot c < b \cdot c$
3. Negative Multiplication/Division Rule: Multiplying or dividing by a negative number $c < 0$ REVERSES the direction of the inequality sign:
   $a < b \implies a \cdot (-c) > b \cdot (-c)$

Example: $-2x \le 8 \implies x \ge -4$ (sign flipped after dividing by $-2$).

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Estimation Techniques & GRE Benchmarks

On the GRE, exact arithmetic is often unnecessary when evaluating choices or placing values in Quantitative Comparison questions. Strategic rounding and benchmark approximations save substantial time.

Essential Benchmarks to Memorize

Rounding & Fast Bounding Example

Estimate $\frac{19.8 \times 40.2}{7.9}$:

• Round terms to friendly benchmark integers: $\frac{20 \times 40}{8} = \frac{800}{8} = 100$.
• Since $19.8 < 20$, $40.2 > 40$, and $7.9 < 8$, the estimation is extremely close to the exact value ($100.75$).

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Bounding Quantities & Upper/Lower Bound Traps

When variables $x$ and $y$ are restricted to intervals (e.g., $a \le x \le b$ and $c \le y \le d$), determine extreme bounds for combinations ($x + y, x - y, x \cdot y, \frac{x}{y}$) by testing boundary endpoints.

Interval Arithmetic Bounds Rules

• Maximum of $(x - y)$: $\max(x) - \min(y)$
• Minimum of $(x - y)$: $\min(x) - \max(y)$
• Maximum of $(x \cdot y)$: $\max(a \cdot c, a \cdot d, b \cdot c, b \cdot d)$
• Minimum of $(x \cdot y)$: $\min(a \cdot c, a \cdot d, b \cdot c, b \cdot d)$

Worked Example: Bounds Calculation

If $-4 \le x \le 3$ and $2 \le y \le 6$, find the maximum possible value of $x^2 - y$:

1. Find bounds for $x^2$: Since $x \in [-4, 3]$, $x^2$ ranges from $0^2 = 0$ (at $x=0$) to $(-4)^2 = 16$. So $0 \le x^2 \le 16$.
2. Maximize $x^2 - y$: Take $\max(x^2) - \min(y) = 16 - 2 = 14$.

• Maximum Value: $14$