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Absolute Value
and Order

Master absolute value, distance on the number line, and ordering real numbers: the foundational skills tested on the GRE Quantitative section.

GRE Quantitative

What Is Absolute Value?

The absolute value of a number \(x\), written \(|x|\), is its distance from zero on the number line. Distance is always non-negative, so \(|x| \geq 0\) for every real number \(x\).

\(|7| = 7\)
7 is already positive: distance from 0 is 7.
\(|-7| = 7\)
\(-7\) is 7 units left of 0: distance is still 7.

Distance on the Number Line

-5 -3 -1 0 1 3 5 -3 3 units 3 3 units

Why Absolute Value Matters on the GRE

The GRE Quantitative section tests your ability to compare quantities, order numbers, and evaluate expressions involving absolute value. These concepts appear in both Quantitative Comparison and Problem Solving questions.

Comparison
Is \(|a - b|\) greater than, less than, or equal to \(|a| - |b|\)?
Ordering
Rank \(-4,\; |{-3}|,\; 2,\; -|{5}|\) from least to greatest.
Equations
Solve \(|2x - 1| = 7\) and determine the solution set.

Core Properties of Absolute Value

1
Non-negativity: \(|x| \geq 0\) for all \(x\), and \(|x| = 0\) only when \(x = 0\).
2
Symmetry: \(|-x| = |x|\). The sign does not affect distance.
3
Multiplicativity: \(|ab| = |a| \cdot |b|\). Absolute value distributes over products.
4
Triangle Inequality: \(|a + b| \leq |a| + |b|\). Equality holds when \(a\) and \(b\) share the same sign.

The Triangle Inequality Visualized

Same sign: \(a = 3, b = 4\) 0 3 7 a = 3 b = 4 |a+b| = 7 = |a|+|b| (equality!) Opposite signs: \(a = 3, b = -4\) 0 3 -4 -1 |a+b| = 1 < 7 = |a|+|b| (strict!)

Absolute Value vs. Signed Value

Signed Value Absolute Value
Definition Position on number line Distance from zero
Can be negative? Yes Never
\(x = -5\) \(-5\) \(5\)
Ordering rule Left is less on number line Closer to 0 is less
GRE trap \(-3 < -1\) \(|-3| > |-1|\)

Ordering Real Numbers

The order of real numbers is determined by their position on the number line: further right means greater. Key rules:

1
Every positive number is greater than every negative number.
2
Among negatives: the one closer to zero is greater. Example: \(-2 > -7\).
3
Zero is greater than every negative number and less than every positive number.
4
Absolute value preserves order for positives: if \(0 < a < b\) then \(|a| < |b|\).

How to Solve Absolute Value Problems

1
Isolate the absolute value expression on one side of the equation or inequality.
2
Split into cases. For \(|E| = k\): write \(E = k\) or \(E = -k\). For \(|E| < k\): write \(-k < E < k\).
3
Solve each case separately using standard algebra.
4
Check all solutions by substituting back into the original expression.
5
Compare or order the results as the question requires.

Example 1: Solving \(|2x - 5| = 9\)

1
Case A: \(2x - 5 = 9 \;\Rightarrow\; 2x = 14 \;\Rightarrow\; x = 7\)
2
Case B: \(2x - 5 = -9 \;\Rightarrow\; 2x = -4 \;\Rightarrow\; x = -2\)
3
Verify: \(|2(7) - 5| = |9| = 9\) ✓    \(|2(-2) - 5| = |-9| = 9\) ✓

Example 2: Solving \(|x + 3| < 5\)

Strategy: \(|E| < k\) becomes \(-k < E < k\).

1
Rewrite: \(-5 < x + 3 < 5\)
2
Subtract 3: \(-8 < x < 2\)
-8 -3 2 center

Example 3: Ordering Expressions

Problem: Arrange from least to greatest: \(-6,\; |-4|,\; 3,\; -|5|,\; |2|\)

1
Evaluate: \(|-4| = 4\), \(-|5| = -5\), \(|2| = 2\)
2
List: \(-6,\; 4,\; 3,\; -5,\; 2\)
3
Order: \(-6 < -5 < 2 < 3 < 4\)
-6 -5 0 2 3 4

Distance Between Two Points

The distance between two numbers \(a\) and \(b\) on the number line is \(|a - b|\). This is always non-negative and order does not matter: \(|a - b| = |b - a|\).

a = -2 b = 5 0 |a - b| = 7

GRE Quantitative Comparison Strategy

When comparing expressions with absolute value, always test with positive, negative, and zero values. The relationship may change depending on the sign.

Test \(x = 3\)
\(|x| = 3\), \(x = 3\). Quantity A = Quantity B.
Test \(x = -3\)
\(|x| = 3\), \(x = -3\). Quantity A > Quantity B.

Common Mistakes to Avoid

✕ Dropping the negative
Assuming \(|x| = x\) always. It equals \(-x\) when \(x < 0\).
✕ Distributing over sums
Writing \(|a + b| = |a| + |b|\). This is only true when \(a\) and \(b\) share signs.
✕ Forgetting the second case
Solving only \(E = k\) and missing \(E = -k\). Always split into two cases.

Quick Reference Table

Expression Meaning Equivalent Form
\(|x| = k\) Distance from 0 is \(k\) \(x = k\) or \(x = -k\)
\(|x| < k\) Within \(k\) of 0 \(-k < x < k\)
\(|x| > k\) More than \(k\) from 0 \(x < -k\) or \(x > k\)
\(|a - b|\) Distance between \(a\) and \(b\) Always \(\geq 0\)
\(|ab|\) Product of distances \(|a| \cdot |b|\)

Question 1 of 5

What is the value of \(|{-12}| + |{3}|\)?

Question 2 of 5

If \(|x - 4| = 7\), which of the following is a possible value of \(x\)?

Question 3 of 5

Which of the following is always true for all real numbers \(a\) and \(b\)?

Question 4 of 5

The solution set of \(|2x + 1| < 5\) is:

Question 5 of 5

Arrange from least to greatest: \(-|{8}|,\; |{-3}|,\; -2,\; |{1}|\)