Absolute Value Inequality Calculator
This absolute value inequality calculator takes the inequality whole — bars, coefficient, constant and all — and gives the split it turns into, the answer in words and in interval notation, and a number line.
Type |x − 3| < 5 and you get −2 < x < 8. Type |x − 3| > 5 and you get two pieces: x < −2 or x > 8.
Change that symbol to an = and you are asking a different question with a different shape of answer — two points rather than a range — which is what the absolute value equation calculator is for.
What the Absolute Value Inequality Calculator Does Not Ask For
Every other tool for this job wants the parts separately. One asks for the coefficients a, b, c and d of the form a|bx+c|+d. Another asks you to type what goes inside the bars, then click one of four symbol buttons, then enter the number on the right.
Type it the way it is printed in the book. Nothing is chosen before you start.
The Bars Mean Distance
|x − 3| is how far x is from 3. That single reading explains everything that follows.
Distance has no direction, which is why one inequality turns into two. And a distance is never negative, which is why the number on the right can settle the whole question before any splitting happens.
Get the Bars Alone First
2|x + 3| − 1 > 5 is not ready to split. Move the −1 across, then divide by the 2, and you have |x + 3| > 3.
If the number multiplying the bars is negative, the symbol turns round at that division — and it turns before the split, not after. Doing those two in the wrong order is where a large share of wrong answers begin. The turn itself is the ordinary rule from solving linear inequalities; only its timing is particular to this page.
Less Than Traps It, Greater Than Throws It Out
|A| < k → −k < A < k — an and, one unbroken range
|A| > k → A < −k or A > k — an or, two pieces
Staying within a distance means being on both sides of the middle at the same time, so both conditions have to hold. Staying further away than a distance can happen in either direction, so one of them is enough — and the answer has a gap in the middle where the near numbers used to be.
What comes out of the split is an ordinary compound inequality, and from that point on the bars have nothing more to do with it.
A Negative Number on the Right Ends It
|x − 3| < −2 has no solution. No distance is less than a negative number.
|x − 3| > −2 is true for every number, for the same reason from the other side. Neither one needs splitting, and splitting them anyway produces answers that look reasonable and are wrong.
Zero on the Right Splits Four Ways
This is the case most pages skip, and the two in the middle look exactly like the negative ones while behaving nothing like them.
|A| < 0 → no solution — no distance is below zero
|A| ≤ 0 → exactly one point, where A is zero
|A| > 0 → every number except that point — two pieces
|A| ≥ 0 → every number
So |x − 3| ≤ 0 has the single answer x = 3, which is not a range at all, and |x − 3| > 0 is everything with a hole punched at 3. Those two are worth recognising on sight.
When the Answer Is One Number
An inequality that turns out to have exactly one solution surprises people, and it is usually assumed to be a mistake.
It is not. |2x + 6| ≤ 0 asks for a distance that is at most zero, and the only distance that qualifies is zero itself — which happens where 2x + 6 = 0, at x = −3. Written properly it is {−3}, a set with one member, not an interval.
Writing the Answer Down
A square bracket means the end is included, a round one means it is not, and infinity always gets a round one.
−2 < x < 8 is (−2, 8). Two pieces are joined with the union sign: (−∞, −2) ∪ (8, ∞). A single point is written in braces, {3}, and no solution at all is ∅.
Test With the Bars, Not With Your Split
Checking your answer against the two inequalities you split it into only proves the arithmetic after the split was right. It cannot catch a split that was wrong.
Put the number back into the original, work the distance out, and compare. That is the only test that catches a symbol which should have turned and did not — and it is what this page does at every endpoint, and a thousandth either side of it, before showing you anything.
Using the Absolute Value Inequality Calculator
Type the inequality as it stands. The bars may sit on either side, you can write abs(x − 3) if bars are awkward on your keyboard, and a coefficient or a constant outside the bars is fine. You get the isolated form, the sign of what is left on the right and what that decides, the split where there is one, the answer in words and in interval notation, and a number line with the distance drawn on it.
Absolute Value Inequality Calculator FAQ
How do you solve an absolute value inequality?
When is it and, and when is it or?
What if the right-hand side is negative?
What happens when the right-hand side is zero?
Can an absolute value inequality have exactly one answer?
Isolate the bars, read the sign of what is left, and only then split.