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Absolute Value Inequalities Calculator

Math

Solve absolute value inequalities of the form |ax + b| < c and find the resulting range of x.

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0.011,000,000
-1,000,0001,000,000
01,000,000

Lower Bound

-5.0000

The lower end of the solution range.

Upper Bound

2.0000

The upper end of the solution range.

Solution Range

-5 < x < 2

The full solution written as a compound inequality.

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Odeh Ahwal

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Odeh Ahwal
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Frequently Asked Questions

How is an absolute value inequality solved?

|ax+b| < c means the expression ax+b is less than c units from zero in either direction, which unfolds into the compound inequality -c < ax+b < c. Solving all three parts for x gives the solution range.

Why does this calculator assume the coefficient a is positive?

Dividing an inequality by a negative number flips the inequality signs, which this calculator does not handle automatically. For a positive coefficient a, the direction of the inequality stays the same throughout, keeping the solution as a single, simple range.

What would happen with a "greater than" inequality instead?

|ax+b| > c has a different shape of solution: instead of one connected range, it splits into two separate rays, x < (-c-b)/a or x > (c-b)/a, since being more than c units from zero can happen in either direction.

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