Solve inequalities of the form |ax + b| < c, <= c, > c, or >= c
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b =
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c =
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Solution Set
Step-by-Step Derivation
Absolute Value Inequality Formula
|u| < c → -c < u < c; |u| > c → u < -c or u > c
Absolute value inequalities turn distance limits into intervals. Less-than cases stay between -c and c, while greater-than cases fall outside those boundaries. Zero and negative c values need special checks.
⚠Note: This solver works with real-number linear absolute value inequalities. It reports empty sets, all real numbers, single-point cases, open intervals, closed intervals, and outside intervals.
How Absolute Value Inequalities Work
The solver checks boundary cases first, then converts the inequality into inside or outside intervals.
Inside Case
Less-than signs keep ax+b between -c and c.
Outside Case
Greater-than signs split into two separate rays.
Endpoint Type
Strict signs use open endpoints; inclusive signs use closed endpoints.
Boundary Check
Zero and negative c values must be handled before splitting.
💡 Example: |2x-3| < 5 becomes -5 < 2x-3 < 5, so -2 < x < 4.
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