< or <=
Inside the interval around the center.
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GMAT Quant
Absolute Value Inequalities
Convert absolute-value inequalities into inside or outside intervals and use midpoint-radius form.
Pattern 1
Small absolute value means inside a band.
If |A| < B, then -B < A < B. With <=, include both endpoints.
Example: |x - 8| < 3
- Lower bound
- 8 - 3 = 5
- Upper bound
- 8 + 3 = 11
- Interval
- 5 < x < 11
Pattern 2
Large absolute value means outside a band.
If |A| > B, then A > B OR A < -B. With >=, include the boundary values.
Example: |x - 8| > 3
- Left region
- x < 5
- Right region
- x > 11
- Meaning
- More than 3 units from 8
The inequality sign tells you the geometry.
> or >=
Outside the interval, in two directions.
Midpoint method
Intervals can be rewritten as midpoint plus radius.
For a < x < b, midpoint m = (a + b)/2 and radius r = (b - a)/2, so |x - m| < r.
Example: 4 < x < 18
- Midpoint
- 11
- Radius
- 7
- Absolute-value form
- |x - 11| < 7
Mini application
Try this: rewrite |x - 5| >= 6 as ordinary inequalities.
Decide whether the answer is one interval or two regions.
Solution
The result is x <= -1 or x >= 11.
The distance from 5 must be at least 6, so x lies on or beyond either boundary.
Remember this
Think distance first, algebra second.
"Close to" a center means inside; "far from" a center means outside.