Inequality Calculator

Fill three boxes, click Calculate. This inequality calculator handles any one-variable inequality of the form $ax + b < c$. The same form with $>$, $\le$ or $\ge$ works too.

• Coefficient of x (a) — the number multiplying $x$, such as 2 in $2x + 3 < 11$
• Constant (b) — the number added to $ax$, such as 3
• Right-hand side (c) — what the left side is compared against, such as 11

You get two rows back. The boundary point solves the equation $ax + b = c$. It is where the answer starts or ends. The probe value is the left side computed one unit right of that boundary. Compare the probe with $c$. If the probe satisfies your inequality, solutions run to the right; if not, they run left.

This page is more than a static inequality calculator — it is also an AI tutor you can ask. The question box takes plain words, photos, and follow-ups. Ask it to fill the three boxes from a word problem. Ask why dividing by a negative flips the sign. Every number still comes from the form on this page, so nothing is guessed.

An equation hands you a handful of answers. An inequality hands you a whole stretch of them. That stretch is called the solution set. Take $2x + 3 < 11$: the answer is every $x$ below 4, not one single number. A search for a solve inequality calculator is really a search for the two ends of that stretch. Most hunts for solutions for inequality problems are the same hunt for two ends. The inequality calculator graph below shows the answer at a glance.

Read the picture: the open dot at 4 says that 4 itself is out. The shading covers every number less than 4, and each shaded point is one of the solutions to the inequality. Ask what is solution of an inequality, and the answer is exactly this shaded stretch. In interval notation the same answer reads $(-\infty, 4)$. So the inequality calculator reports both forms. It works as an inequality solver, and it doubles as an interval notation calculator.

Here is what separates an inequality calculator from an equation solver: the flip rule. Add or subtract any number from both sides, and the direction never changes. Multiply or divide by a positive number, and it still never changes. Multiply or divide by a negative number, and the sign must reverse: $<$ becomes $>$, $\le$ becomes $\ge$.

This is an old observation. Smith and Gale's Elements of Analytic Geometry (1904) put it in one line: “changing signs throughout an inequality reverses the inequality sign.” People sometimes call the result inequality equations — but a true inequality keeps its sign, or flips it whole.

The letters on this page match the form fields exactly:

The boundary sits where $ax + b = c$, so $x = \dfrac{c - b}{a}$. That is the first row the inequality calculator returns. On the second row, the inequality calculator applies the flip rule for you.

Type any $a$, $b$, $c$ and the inequality calculator answers in two rows. Row one is the boundary: the exact point where $ax + b = c$. Row two is the probe: the inequality calculator drops one point past the boundary and reports the left side there. Compare the probe with $c$, and the direction of the solution set is decided — no guesswork, no sign errors. The inequality calculator therefore answers two questions at once: where the edge is, and which side is in.

Solve $2x + 3 < 11$, then describe the solution set.

1. Subtract 3 from both sides: $2x < 8$.
2. Divide by 2 — positive, so no flip: $x < 4$.
3. Boundary point 4, open dot, shading to the left. Interval notation: $(-\infty, 4)$.

Type $a = 2$, $b = 3$, $c = 11$ into the inequality calculator above. It returns the boundary 4 and the probe value 13. At one unit right of 4 the left side reads $13$, and $13 < 11$ is false. The solutions therefore lie to the left — exactly where the inequality calculator pointed. Confirm the interval $(-\infty, 4)$ with the inequality calculator in the same run.

Solve $-3x + 2 \ge 11$, then describe the solution set.

1. Subtract 2 from both sides: $-3x \ge 9$.
2. Divide by $-3$ — negative, so flip: $x \le -3$.
3. Boundary point $-3$, closed dot, shading to the left. Interval notation: $(-\infty, -3]$ — read it as the interval the inequality calculator writes beside the graph.

Set $a = -3$, $b = 2$, $c = 11$ in the inequality calculator. It returns boundary $-3$ and probe value $8$. One unit right of $-3$ the left side reads $8$, and $8 \ge 11$ is false, so the solutions run left. Because $a$ is negative, the inequality calculator's flip rule turns that into $x \le -3$. The picture below agrees.

Read it backwards too: with $a = -3$, $b = 2$ and boundary $-3$, the right side is $c = (-3)(-3) + 2 = 11$ — exactly the $c$ you started with. Any two of $a$, $b$, $c$ and the boundary pin down the fourth.

Same shape, different signs — watch the inequality calculator handle each case. $2x + 3 < 11$ has $a = 2$: positive, no flip, answer $x < 4$. $-3x + 2 \ge 11$ has $a = -3$: negative, flip, answer $x \le -3$. And $4x - 7 \ge 1$ has $a = 4$: positive again, so the inequality calculator keeps the direction and returns $x \ge 2$. Run all three through the inequality calculator and compare the rows: the boundary moves with $b$ and $c$, while the flip rule follows the sign of $a$.

1. Forgetting the flip. Dividing $-3x \ge 9$ by $-3$ and writing $x \ge -3$ keeps the wrong direction — the answer is $x \le -3$. Test $x = -4$: it works in the original, and $-4 \ge -3$ is false.
2. Mismatched brackets in interval notation. A strict inequality takes a parenthesis, an inclusive one takes a square bracket. Infinity always gets a parenthesis, because you never reach it.
3. Testing the boundary instead of the inside. The boundary is where the answer changes sides, so it usually fails the test. Pick an interior point, such as $x = 0$ for $x < 4$ — or read the probe row, which is the inequality calculator testing that point for you.

The inequality calculator never skips the flip. On paper, the test point is your safety net: one quick check turns a wrong inequality solution into a right one. All three mistakes are one-glance fixes on the inequality calculator's two rows.