Inequality Solver | Linear & Quadratic
Solve linear and quadratic inequalities with step-by-step solutions.
What Is the Inequality Solver | Linear & Quadratic?
The Linear Inequality Solver finds all values of x satisfying ax + b OP c, where OP is any of <, ≤, >, or ≥. Unlike an equation (which has one solution), an inequality describes an entire region, a ray or half-line on the number line. The solution is expressed in both inequality form and interval notation.
- ›Four operators: supports strict (<, >) and non-strict (≤, ≥) comparisons; brackets vs parentheses in interval notation reflect the distinction.
- ›Automatic flip rule: when a is negative, the direction automatically reverses, the most common source of errors in manual solving.
- ›Special cases: handles a = 0 (either always true or always false), returning "all real numbers" or "no solution" with explanation.
- ›Number line diagram: visual representation with open/closed dots and directional shading.
- ›Step-by-step working: each arithmetic step shown, subtract b, divide by a, flip if needed.
- ›Preset examples: one-click loading of common textbook problems.
Formula
| Operator | Solution x | Interval Notation | Number Line |
|---|---|---|---|
| x < k | x < k | (−∞, k) | Open circle at k, arrow left |
| x ≤ k | x ≤ k | (−∞, k] | Closed circle at k, arrow left |
| x > k | x > k | (k, +∞) | Open circle at k, arrow right |
| x ≥ k | x ≥ k | [k, +∞) | Closed circle at k, arrow right |
How to Use
- 1Enter the coefficient a (the number multiplying x). It can be any real number, including negative.
- 2Enter the constant b (added or subtracted on the left side). Use negative values to model subtraction.
- 3Enter the right-hand side value c.
- 4Select the comparison operator (< ≤ > ≥) from the dropdown.
- 5Press Solve (or Enter) to see the solution, step-by-step working, interval notation, and number line.
- 6Use the preset buttons to load common examples instantly.
- 7Press Reset (or Escape) to clear all fields.
Example Calculation
Example 1, Negative coefficient (flip required)
Solve: −3x + 6 > −9
Example 2, Non-strict inequality
Solve: 2x − 4 ≤ 8
Example 3, Degenerate case
Solve: 0·x + 3 < 7
Understanding Inequality | Linear & Quadratic
Why Inequalities Are Different from Equations
An equation like 2x + 3 = 7 has exactly one solution (x = 2). A linear inequality like 2x + 3 < 7 has infinitely many solutions, all x less than 2. This is because an inequality defines a condition: the set of all inputs that make the statement true. On a number line, this always produces a ray (half of the line), and on a coordinate plane, it produces a half-plane.
The defining rule that distinguishes inequality solving from equation solving is the flip rule for negative multipliers. When you multiply or divide both sides of an inequality by a negative number, the direction reverses. This is because multiplying by −1 reflects both sides across zero on the number line, swapping their relative order. For example: 3 > 1, but −3 < −1.
Interval Notation Explained
Interval notation is a compact way to describe sets of real numbers using brackets and parentheses:
- ›Parenthesis ( or ) means the endpoint is NOT included (strict inequality < or >).
- ›Bracket [ or ] means the endpoint IS included (non-strict inequality ≤ or ≥).
- ›Infinity (∞ or −∞) is always written with a parenthesis, infinity is not a number and cannot be "included."
- ›All real numbers is written (−∞, +∞); no solution is written as the empty set ∅.
Where Linear Inequalities Appear in Practice
- ›Budget constraints: total spending ≤ budget; each item has a cost × quantity ≤ total available.
- ›Safety limits: temperature, pressure, or load must remain within acceptable bounds.
- ›Dosage ranges: medication concentration must stay between a minimum effective level and a maximum safe level.
- ›Physics constraints: speed must remain below the speed of sound in subsonic flight design.
- ›Linear programming: the foundation of optimization in operations research, objective function is maximized/minimized subject to a system of linear inequalities called constraints.
Compound Inequalities
A compound inequality like 1 < 2x + 3 < 9 combines two inequalities simultaneously. To solve, treat each half separately:
- ›Left part: 1 < 2x + 3 → 2x > −2 → x > −1
- ›Right part: 2x + 3 < 9 → 2x < 6 → x < 3
- ›Combined: −1 < x < 3, or interval (−1, 3)
This calculator solves single linear inequalities ax + b OP c. For compound inequalities, solve each part using this tool and intersect the resulting intervals manually.
Frequently Asked Questions
Why does the inequality direction flip when dividing by a negative number?
Multiplying both sides by −1 swaps their positions on the number line. Consider 2 > 1: true. Multiply both sides by −1: you get −2 and −1. But −2 < −1, the order reversed. The same logic applies to any negative multiplier or divisor.
If you forget this rule, you get the exact opposite answer, a very common algebra mistake. A good self-check: plug a value from your solution back into the original inequality. If it doesn't satisfy it, you forgot to flip the sign.
What is the difference between strict and non-strict inequalities?
A strict inequality (<, >) excludes the boundary value, the endpoint is NOT part of the solution set, shown as an open circle on the number line and a parenthesis in interval notation.
A non-strict inequality (≤, ≥) includes the boundary value, shown as a closed (filled) circle and a square bracket. For example, x < 5 means all numbers less than 5 but NOT 5 itself; x ≤ 5 includes 5 as a valid solution.
What happens when a = 0?
When a = 0, x disappears from the inequality, leaving just b OP c, a statement that is entirely independent of x.
- ›If the statement is always true (e.g., 3 < 7), every real number is a solution: (−∞, +∞).
- ›If the statement is always false (e.g., 7 < 3), there is no solution: ∅.
This degenerate case frequently trips up students when simplifying more complex expressions. The calculator detects a = 0 automatically and explains which case applies.
How do I solve a compound inequality like −1 ≤ 3x − 2 < 10?
Treat all three parts simultaneously by performing the same operation on every section. For −1 ≤ 3x − 2 < 10:
- ›Add 2 to all three parts: 1 ≤ 3x < 12
- ›Divide all by 3: 1/3 ≤ x < 4
- ›Result in interval notation: [1/3, 4)
If you ever need to divide by a negative number, reverse BOTH inequality signs simultaneously. This calculator handles single inequalities; use the technique above manually for compound ones, or apply this tool twice and intersect the intervals.
Can this solve inequalities with x on both sides like 3x + 1 > x + 5?
Not directly, this solver requires the form ax + b OP c with x only on the left. However, you can rearrange any two-sided inequality first:
- ›3x + 1 > x + 5
- ›Subtract x from both sides: 2x + 1 > 5
- ›Now set a = 2, b = 1, c = 5, operator > and solve
The same rearrangement technique converts any linear inequality with x on both sides into the ax + b OP c form. Always collect x terms on the left before entering values into this tool.
What does interval notation look like for "no solution" and "all real numbers"?
"No solution" is written as ∅ (the empty set symbol), no real number satisfies the inequality. "All real numbers" is written as (−∞, +∞) or ℝ.
Both cases arise when a = 0, leaving a constant comparison. They also arise from contradictory compound inequalities, for example, x < 2 AND x > 5 simultaneously has no solution because no number is both less than 2 and greater than 5.
How are linear inequalities used in linear programming?
Linear programming (LP) optimizes a linear objective function, maximize profit, minimize cost, subject to constraints expressed as linear inequalities. Each constraint defines a half-plane in 2D or a half-space in higher dimensions.
The feasible region (all solutions satisfying every constraint simultaneously) is always a convex polygon. The optimal solution always occurs at a vertex (corner point) of this feasible region, which is why the simplex algorithm works by jumping between vertices.
LP underlies supply chain optimization, portfolio allocation, transportation planning, and scheduling in virtually every industry that manages constrained resources.