Inequality Solver — Linear & Quadratic
Solve linear and quadratic inequalities with step-by-step solutions.
Solve ax + b <op> c (linear inequality)
What Is the Inequality Solver — Linear & Quadratic?
The Linear Inequality Solver finds the set of all values of x that satisfy an inequality of the form ax + b OP c, where OP is one of <, ≤, >, or ≥. The key rule that distinguishes solving inequalities from equations is that multiplying or dividing both sides by a negative number reverses the inequality direction.
Formula
How to Use
Enter a, b, and c for the inequality ax + b OP c, and select the comparison operator (< ≤ > ≥) from the dropdown. Click Solve to see the solution, the direction flip rule (if applicable), and the answer in both inequality and interval notation.
Example Calculation
Solve: −3x + 6 > −9 (a=−3, b=6, c=−9, OP = >) Step 1: −3x > −9 − 6 = −15 Step 2: x < −15/−3 = 5 ← direction flips (dividing by −3) Solution: x < 5, interval: (−∞, 5) Solve: 2x − 4 ≤ 8 2x ≤ 12 → x ≤ 6, interval: (−∞, 6]
Understanding Inequality — Linear & Quadratic
Linear inequalities are closely related to linear equations but describe regions rather than points. On a number line, the solution to a linear inequality is always a ray or the entire line. On a coordinate plane, it is a half-plane.
Inequalities appear throughout optimization: budget constraints (spending ≤ budget), safety limits (temperature ≥ minimum), and dosage ranges (concentration must stay between bounds). Linear programming, the foundation of operations research, is entirely based on systems of linear inequalities.
The flip rule for negative coefficients is one of the most commonly forgotten facts in algebra. A helpful mnemonic: the "hungry alligator" symbol always points toward the larger value, so multiplying by a negative reverses which side is larger.
Frequently Asked Questions
Why does the inequality flip when dividing by a negative?
Consider 2 > 1. Multiplying both sides by −1 gives −2 and −1; since −2 < −1, the direction must flip to maintain truth. This holds for any negative multiplier or divisor.
What is interval notation?
Interval notation uses parentheses ( ) for strict inequalities (< or >) and brackets [ ] for non-strict inequalities (≤ or ≥). Infinity is always written with a parenthesis since it is not a real number.
What if a = 0?
If a = 0, the inequality becomes b OP c, which is either always true (e.g., 3 < 7) giving all real numbers as solutions, or always false (e.g., 7 < 3) giving no solution.
Can this solve compound inequalities like 1 < 2x+3 < 9?
Not directly. For compound inequalities, solve each part separately and intersect the solution sets.
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