Linear Inequalities (Inecuaciones) Calculator
Use this tool to solve inequalities in the form ax + b < c, ax + b ≤ c, ax + b > c, or ax + b ≥ c.
What Is an Inecuación?
An inecuación is the Spanish term for an inequality. Instead of saying two expressions are exactly equal, an inequality compares them using symbols like <, >, ≤, and ≥.
For example, the equation 2x + 3 = 15 has one exact solution. But the inequality 2x + 3 < 15 has a range of valid solutions.
How This Inequalities Calculator Works
Standard form used
The calculator solves expressions of this form:
- ax + b < c
- ax + b ≤ c
- ax + b > c
- ax + b ≥ c
Method
It follows the same classroom method:
- Move constants to isolate the variable term.
- Divide by the coefficient of x.
- If dividing by a negative number, flip the inequality sign.
- Return both symbolic and interval notation results.
Why Flipping the Sign Matters
If you multiply or divide both sides of an inequality by a negative number, the order reverses. That is why:
- x < 5 becomes -x > -5
- 3x ≥ 12 gives x ≥ 4
- -3x ≥ 12 gives x ≤ -4
This calculator applies that rule automatically, helping avoid one of the most common mistakes in algebra.
Examples You Can Try
Example 1: Positive coefficient
Enter: a = 2, b = 3, sign <, and c = 15. You should get x < 6, with interval (-∞, 6).
Example 2: Negative coefficient
Enter: a = -4, b = 8, sign ≥, and c = 0. Rearranging gives -4x ≥ -8, then dividing by -4 flips the sign to x ≤ 2.
Common Student Errors (and How to Avoid Them)
- Forgetting to flip the sign when dividing by a negative number.
- Moving constants across the inequality with wrong signs.
- Confusing < with ≤, especially in interval notation.
- Stopping at symbolic form and not checking on a number line.
Who This Tool Is For
This inecuaciones calculator is useful for:
- Middle school and high school algebra students.
- Test prep for SAT/ACT and placement exams.
- Teachers who want quick answer checks during class.
- Parents helping with homework.
Final Note
A good inequalities solver should do more than output an answer; it should show structure and logic. Use the calculator above, then review the steps so you can solve inecuaciones confidently by hand as well.