Calculator
Example Data Table
| Type | Input | Expected Output |
|---|---|---|
| Evaluate | |2x - 3| at x = 5 | 7 |
| Single inequality | |x - 4| ≤ 2 | [2, 6] |
| Range inequality | 1 < |x + 1| ≤ 4 | [-5, -2) ∪ (0, 3] |
| Compound linear | 2 ≤ 3x + 1 < 10 | [0.3333333333, 3) |
Formula Used
Absolute value rule: |u| equals u when u is positive or zero. It equals -u when u is negative.
Centered form: |u| < k becomes -k < u < k. Also, |u| ≤ k becomes -k ≤ u ≤ k.
Outer form: |u| > k becomes u < -k or u > k. Also, |u| ≥ k becomes u ≤ -k or u ≥ k.
Linear step: For ax + b compared with c, subtract b first. Then divide by a. Reverse the sign when dividing by a negative number.
Interval notation: Use parentheses for open endpoints. Use brackets for closed endpoints.
How to Use This Calculator
- Select a calculation mode.
- Enter the coefficient a and constant b.
- Fill only the fields shown for the chosen mode.
- Pick the needed inequality signs.
- Press Calculate to view the result above the form.
- Review the interval notation and step summary.
- Download a CSV file or PDF report when needed.
Absolute Value and Inequalities Guide
Why this topic matters
Absolute value and inequalities appear in algebra, graphs, and exam practice. They measure distance from zero. They also describe ranges, limits, and allowed values. A focused calculator saves time. It reduces sign errors. It also helps students see why an answer is an interval.
What this calculator solves
This tool handles four common tasks. It evaluates |ax + b| for any input. It solves a single absolute inequality such as |2x - 5| ≤ 7. It solves bounded forms like 1 < |x + 3| ≤ 6. It also solves compound linear inequalities without absolute value.
How the logic works
Absolute value is about distance. When the answer must stay below a bound, the solution stays between two numbers. When the answer must stay above a bound, the solution moves outside two numbers. That is why some results show one interval and others show a union of intervals.
Why interval notation helps
Interval notation is compact and clear. Parentheses mean an endpoint is not included. Brackets mean it is included. Students can compare answers fast. Teachers can check work faster. Tutors can also explain the change from strict to inclusive signs with less confusion.
Where students make mistakes
The most common mistake happens after dividing by a negative number. The inequality sign must reverse. Another common issue is mixing up “and” with “or.” Inner regions usually use “and.” Outer regions usually use “or.” This calculator highlights both patterns in the step summary.
Best way to study with it
Start with easy integers. Then try fractions and negative coefficients. Compare manual work with the calculator output. Use the example table as a model. Export results for notes or revision sheets. Repeating that process builds stronger algebra habits and better confidence before quizzes.
Frequently Asked Questions
1. What does absolute value mean?
Absolute value shows distance from zero on the number line. It never gives a negative result. For example, |-5| and |5| both equal 5.
2. Why do some answers use two intervals?
That happens when the solution sits outside a center region. Inequalities like |u| > k split into two parts. One lies left of -k, and the other lies right of k.
3. When do I reverse the inequality sign?
Reverse the sign only when you divide or multiply by a negative number. This step is easy to miss. It changes the direction of the solution set.
4. What is the difference between < and ≤?
The symbol < excludes the endpoint. The symbol ≤ includes it. In interval notation, open endpoints use parentheses, while included endpoints use brackets.
5. Can this tool handle negative constants on the right side?
Yes. The tool checks those cases. Some inequalities become impossible. Others become true for all real numbers, depending on the chosen sign.
6. Why does the result say all real numbers?
That means every real x satisfies the statement. It often appears when an absolute value is compared with a negative number using a greater-than style inequality.
7. Is the calculator useful for homework checking?
Yes. It is useful for checking signs, intervals, and endpoint inclusion. It also helps students compare their handwritten steps with a clean final answer.
8. Can I save the result for later?
Yes. After calculation, use the CSV button for spreadsheet work or the PDF button for a quick printable summary.