What does this calculator do?

This calculator solves 1–3 linear inequalities, combines their solution sets with AND or OR, and displays the final result on a number line.

Each row has the form ax + b [operator] c. The tool isolates x for each inequality, then combines the resulting intervals according to the logical connector you choose.

• Shows the isolate-x step for each inequality.
• Highlights the sign-flip step when dividing by a negative coefficient.
• Draws a number line with open or closed endpoint markers.
• Returns the combined interval notation.
• Displays explicit empty-set and all-real-number results instead of a blank output.

What does it mean to graph an inequality on a number line?

Graphing an inequality on a number line means showing all x-values that satisfy the inequality as a ray, interval, union of intervals, or special set.

For example, x > 2 means all values to the right of 2. The value 2 itself is not included, so the endpoint is open. If the inequality were x ≥ 2, the endpoint would be closed because 2 is included.

How is a linear inequality solved?

A linear inequality is solved by isolating x using the same basic operations as a linear equation, with one extra rule for negative multiplication or division.

When you divide or multiply both sides by a negative number, the inequality direction reverses. This is the most important rule behind many number-line mistakes.

1. Read the inequality in the form ax + b [operator] c.
2. Move b to the other side.
3. Divide by a.
4. If a is negative, reverse the inequality sign.
5. Write the answer on the number line and in interval notation.

What is the difference between AND and OR?

AND means every condition must be true at the same time, so the calculator takes the intersection. OR means at least one condition may be true, so the calculator takes the union.

• AND: x must satisfy the first inequality and the second inequality. The result is the overlapping region.
• OR: x may satisfy either inequality. The result is the combined region.

For example, x > 1 AND x < 5 gives the interval (1, 5). But x < 1 OR x > 5 gives two rays: (-∞, 1) ∪ (5, ∞).

How do open circles, closed circles and interval notation match?

Interval notation is the symbolic version of what the number line shows visually.

• Parentheses ( ) mean the endpoint is not included.
• Brackets [ ] mean the endpoint is included.
• Infinity and negative infinity are never included, so they always use parentheses.
• ∅ means there is no solution; ℝ means all real numbers are solutions.

For example, x ≤ 3 is written as (-∞, 3]. The right endpoint uses a bracket because 3 is included.

Example: graphing one inequality

The inequality 2x - 3 ≥ 5 is solved by isolating x.

1. 2x - 3 ≥ 5
2. 2x ≥ 8
3. x ≥ 4

On the number line, 4 is a closed endpoint and the ray goes to the right. The interval notation is [4, ∞).

Example: sign flip with a negative coefficient

In -3x + 6 < 0, the coefficient of x is negative, so the inequality direction flips in the final division step.

1. -3x + 6 < 0
2. -3x < -6
3. x > 2

The result is x > 2, so 2 is an open endpoint and the ray extends to the right. This shows why the sign-flip rule matters.

Example: using AND to make a bounded interval

If the conditions are x ≥ -2 AND x < 4, the final solution must satisfy both inequalities at once.

The number line has a closed endpoint at -2, an open endpoint at 4, and a shaded segment between them. The interval notation is [-2, 4).

Example: using OR to combine two separate regions

If the conditions are x < -1 OR x ≥ 3, a value only needs to satisfy one of the two inequalities.

The result has two pieces: (-∞, -1) ∪ [3, ∞). The endpoint at -1 is open, the endpoint at 3 is closed, and two separate regions are shaded.

When do empty set and all real numbers appear?

The empty set appears when no x-value can satisfy the combined conditions; all real numbers appear when every x-value is covered by the combined conditions.

For example, x > 5 AND x < 2 is impossible, so the result is ∅. But x < 2 OR x ≥ 2 covers the whole number line, so the result is ℝ.

When is this calculator useful?

This calculator is useful when you want to see the solution, not only read an algebraic answer.

• Checking whether an endpoint should be open or closed.
• Understanding why dividing by a negative changes the inequality direction.
• Comparing AND and OR compound inequalities.
• Learning how interval notation matches a number-line graph.
• Recognizing empty-set or all-real-number results in compound inequalities.

What this calculator does not cover

This calculator is not a general inequality solver; it is focused on linear inequalities on a number line.

• Does not solve absolute value inequalities.
• Does not solve quadratic inequalities.
• Does not build sign charts for rational inequalities.
• Does not shade two-variable inequalities on the coordinate plane.
• Does not accept more than three inequalities at once.

These limits keep the output readable and aligned with the actual number-line model. For a different inequality type, use the more specific calculator.

Number line graph vs interval notation

The number line graph is the visual form of the solution, while interval notation is the compact symbolic form of the same set.

For example, x > 1 AND x ≤ 6 appears as a shaded segment after 1 and up to 6. In interval notation, the same solution is (1, 6].