Calculator Inputs
Enter each linear inequality in the form ax + b compared with a right-side value.
Formula Used
1. Start with each linear inequality in the form ax + b < c, ax + b ≤ c, ax + b > c, or ax + b ≥ c.
2. Move the constant term: ax op (c - b).
3. Divide by a: x op ((c - b) / a).
4. If a is negative, reverse the inequality sign after division.
5. For AND, take the intersection of both solution sets.
6. For OR, take the union of both solution sets.
How to Use This Calculator
- Enter the variable symbol, such as x.
- Fill the first inequality using coefficient, constant, operator, and right-side value.
- Select whether the two inequalities are linked by AND or OR.
- Fill the second inequality in the same format.
- Choose decimal precision and graph padding if needed.
- Press Solve Compound Inequality.
- Read the simplified inequalities, final set, interval notation, and graph.
- Use the CSV or PDF buttons to export the solution summary.
Example Data Table
| Example | Inequality 1 | Join Type | Inequality 2 | Final Solution | Interval Notation |
|---|---|---|---|---|---|
| 1 | 2x + 3 ≤ 11 | AND | x - 1 > 0 | 1 < x ≤ 4 | (1, 4] |
| 2 | 3x - 6 > 9 | OR | x + 2 ≤ 0 | x > 5 or x ≤ -2 | (5, ∞) ∪ (-∞, -2] |
| 3 | -2x + 8 ≥ 0 | AND | x - 6 < 0 | x ≤ 4 | (-∞, 4] |
FAQs
1. What is a compound inequality?
A compound inequality combines two inequalities using AND or OR. AND means both statements must be true together. OR means at least one statement must be true. The final solution is written using interval notation, set notation, or a number line.
2. What is the difference between AND and OR?
AND uses the overlap between the two solution sets. OR combines every value from both sets. In interval language, AND gives an intersection, while OR gives a union. This choice completely changes the final answer.
3. Why does the inequality sign reverse sometimes?
The sign reverses whenever you multiply or divide both sides by a negative number. This preserves the correct order of values. For example, if -2x > 6, dividing by -2 changes the result to x < -3.
4. Can the result ever be one exact point?
Yes. A compound inequality can reduce to one exact value when two boundaries meet and both endpoints are included. For example, x ≥ 3 and x ≤ 3 gives x = 3. The calculator shows that as a valid single-point solution.
5. What does an empty set mean here?
An empty set means no number satisfies the compound inequality. This usually happens with AND cases when the two separate solution ranges do not overlap. The calculator reports it as no solution and shows an empty interval form.
6. Why is interval notation useful?
Interval notation gives a short, standard way to write ranges. Parentheses mark excluded endpoints, while brackets mark included endpoints. It is widely used in algebra, calculus, graphing, and exam solutions because it is compact and precise.
7. Can I use decimals and negative values?
Yes. The calculator accepts decimals, negatives, and zero coefficients. It also handles cases where the variable term disappears, which may lead to all real numbers or no solution depending on whether the remaining statement is true.
8. What does the graph represent?
The graph is a number line of the final solution set. Thick highlighted segments show valid values. Open circles mean endpoints are excluded. Filled circles mean endpoints are included. This makes the answer easier to verify visually.