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QUANTITATIVE REASONING · GMAT-Q-ABS-01
Absolute Value and Distance on the Number Line
GMAT lesson development preview · DRAFT · Original learning and assessment material
Concepts and problem-solving strategy
Absolute-value equation
Inside interval
Outside interval
Sum of distances
Think Like a GMAT Test-Taker
Absolute value measures a nonnegative distance. For |x−c|, the center is c: x can lie on either side of it. Translate the distance condition into points, an inside interval, or two outside intervals before doing algebra. Check any restriction in the original question.
Faster Methods
For |x-c|=r, mark c and move r units in both directions; substitute candidates in the original equation.
For |x-c|<r, write a compound inequality. For |x-c|>r, split into two outside rays.
For sums of distances, use the triangle inequality to find the minimum before testing intervals.
Worked example 1
Solve |x−4|=7.
Seven units to the right of 4 is 11; seven units to the left is −3. Substituting either value gives absolute value 7. Keeping only the positive solution would lose a valid answer.
Worked example 2
Solve |2x−3|<5.
Add 3 throughout and divide by positive 2; strict endpoints stay excluded.
Worked example 3
Find the minimum of |x−6|+|x−10|.
Between 6 and 10, the two distances sum to exactly 4. Outside that interval, moving farther away increases the total. Thus the minimum is 4, attained at every point in the closed interval, not only at its midpoint.
Worked example 4
How many integers satisfy |2x−1|≤7?
Add one to all three parts, then divide by two. The eight integers include both endpoints because the inequality is non-strict.
Common Traps
- Strict inequalities exclude endpoints; non-strict inequalities include them.
- For |A|=k, negative k is impossible; k=0 forces A=0; positive k gives A=k or A=−k. The number of x-values depends on the expression A.
- Squaring can add candidates unless you check the original equation.