Integers & Number Properties • Topic 4 of 5

Absolute Value

The absolute value |x| is the distance of x from 0 on the number line, so it is never negative: |−7| = 7 and |7| = 7. Read it as "how far", not "which side". Two consequences drive most GRE questions. First, |x| = a (with a > 0) has two solutions, x = a and x = −a, so equations with absolute value usually branch. Second, |x − c| is the distance between x and c, which turns range statements into geometry: |x − 4| < 3 means x is within 3 of 4, i.e. 1 < x < 7. Basic rules: |x·y| = |x|·|y| and |x|² = x². The number-one mistake is treating |x| as "make it positive" — for a variable you must consider both signs, and on Quantitative Comparison that second case is often exactly where "cannot be determined" hides.

A number line showing that the absolute value of -3 and the absolute value of 3 are both equal to 3, each three units from zeroAbsolute value = distance from 0-5-4-3-2-1012345|-3| = 3|3| = 3Both -3 and 3 sit 3 units from 0.

✅ Solved examples

1. Solve |x − 2| = 5.
Distance 5 from 2 in both directions: x − 2 = 5 gives x = 7; x − 2 = −5 gives x = −3. Solutions: x = 7 or x = −3.
2. Solve |2x + 1| < 7.
Rewrite as −7 < 2x + 1 < 7. Subtract 1: −8 < 2x < 6. Divide by 2: −4 < x < 3.
3. If |x| = |−4|, what are the possible values of x?
|−4| = 4, so |x| = 4, giving x = 4 or x = −4.
4. Quantitative Comparison. Quantity A: |−6| + |−2|. Quantity B: |−6 + (−2)|. Which is greater?
A = 6 + 2 = 8. B = |−8| = 8. They are equal — a reminder that |a| + |b| equals |a + b| only when a and b share a sign.

✏️ Practice — try these, take hints as needed

1. Solve |x + 3| = 10.
Two cases.
x + 3 = 10 or x + 3 = −10.
Solve each.
x = 7 or x = −13
2. Solve |x − 5| ≤ 2 as a range.
Within 2 of 5.
−2 ≤ x − 5 ≤ 2.
Add 5.
3 ≤ x ≤ 7
3. What is |3 − 9| + |9 − 3|?
|3 − 9| = 6.
|9 − 3| = 6.
Add.
12
4. If |x| = 0, what is x?
Distance 0 from 0.
Only one point.
x = 0.
0
5. How many integer solutions does |x| < 3 have?
−3 < x < 3.
List integers.
−2, −1, 0, 1, 2.
5

📝 Topic test — 8 questions

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