Welcome to CalcuGo's graphing inequalities on a number line calculator, where we'll take on some linear inequalities and see how to plot them on the number line. And once we see how to deal with one, we'll add some more to the pile and get to graphing systems of inequalities. The resulting number line graph is a simple tool to find values that satisfy all given conditions.
🙋 Looking for a different notation? Our inequality to interval notation calculator converts the same solutions into interval and set-builder form.
Linear inequalities
Inequalities in math are numerical relations that describe where a value lies with respect to some other one. By default, there are four of them. They say whether the first number is:
- Smaller than (
<); - Smaller than or equal to (
≤); - Greater than (
>); or - Greater than or equal to (
≥)
the second number. For instance, 123 < 216 states that 123 is smaller than 216.
Such relations seem straightforward when done with numbers. It gets more interesting if we decide to include a bit of algebra and introduce variables. Instead of being specific, inequalities give a rough idea of where the result lies. For instance, x - 10 > 13 states that subtracting 10 from the value of x will always give something greater than 13. As such, inequalities usually give a range of possible results rather than a single one.
Linear inequalities are those with variables only in the first power. They cannot have them in higher exponents, in the denominator of a quotient, under a root, inside a logarithm, etc. Below you can find several examples of linear inequalities:
x ≥ 0-2 < 2x + 74x - 1 ≤ 2x + 5
Here we focus on those with a single variable, which we denote by x. Therefore, we can limit ourselves to one-dimensional drawings and study number line graphs.
How to graph inequalities
We can order all real numbers and mark them on an infinite axis called the number line. In essence, the line tells us where one value lies with respect to others: is it larger (to the right) or smaller (to the left) of something else?
The first lesson on how to graph inequalities is: you have to solve them first. That is, we need to go from expressions like 3x + 1 ≥ 7 to something of the form x ≥ 2 — a single x followed by an inequality relation and a number. We do it the same way we deal with ordinary equations, with the exception that we need to change the inequality sign whenever we multiply or divide by a negative number.
Once we obtain something like x ≥ 2, the basic instructions are:
- Find the value from the inequality on the number line graph.
- Draw the shaded ray from that point to the:
- Left for
<and≤; or - Right for
>and≥.
- Left for
- At the endpoint, draw a small circle and either:
- Keep it empty (open) for strict inequalities (
<and>); or - Fill it (closed) for non-strict inequalities (
≤and≥).
- Keep it empty (open) for strict inequalities (
Our calculator uses this open/closed circle convention. Remember that the number line is infinite in both directions, so the linear inequality plots are also infinite, but in one direction — they mark sets of all numbers from minus infinity to some value (for < and ≤) or from some value to plus infinity (for > and ≥).
| Solved inequality | Endpoint circle | Shaded direction |
|---|---|---|
| x < a | open ○ | left |
| x ≤ a | closed ● | left |
| x > a | open ○ | right |
| x ≥ a | closed ● | right |
Graphing systems of inequalities
Now that we know how to graph a single inequality, our next move is to graph more of them at once. Enter two or three inequalities and choose how to combine them:
- AND (a system) — take the intersection of the solution sets. Only the numbers that satisfy every inequality survive. For example,
x > 2andx ≤ 5gives(2, 5]. If the pieces don't overlap, the system has no solution. - OR (a union) — take the union of the solution sets. A number qualifies if it satisfies at least one inequality. For example,
x < 0orx > 3gives(-∞, 0) ∪ (3, ∞).
Example: using the graphing inequalities on a number line calculator
Suppose we want to graph 3x + 1 ≥ 7. Type it into the first field and press the button. The calculator:
- Solves it: subtract 1 to get
3x ≥ 6, then divide by 3 to getx ≥ 2. - Draws a closed circle at 2 (because of the
≥sign) and shades the ray to the right toward+∞. - Reports the answer in inequality form (
x ≥ 2), interval form ([2, ∞)), and set-builder form ({ x | x ≥ 2 }).
Frequently Asked Questions
How do I graph x ≥ 2 on a number line?
Put a filled (closed) circle at 2, because 2 is included, then shade everything to the right of it toward positive infinity. An arrow shows the ray continues forever.
When do I use an open vs a closed circle?
Use an open (empty) circle for strict inequalities < and > — the endpoint is excluded. Use a closed (filled) circle for ≤ and ≥ — the endpoint is included.
Why does the sign flip when I divide by a negative?
Multiplying or dividing both sides of an inequality by a negative number reverses the order of the two sides, so the inequality symbol must be flipped. For example, -4x < 8 becomes x > -2, not x < -2. The calculator handles this automatically.
How do I graph a system of inequalities?
Enter each inequality in its own field and choose AND to see where all of them overlap (the intersection) or OR to see everywhere at least one holds (the union). The calculator draws each inequality separately and then the combined solution.
What if there is no solution?
If an AND system asks for conditions that can never be true at the same time — for example x < 1 and x > 3 — the intersection is empty, written ∅, and the number line is left blank.