Inequality Calculator

Cum se utilizează acest calculator

1. Introduceți Coefficient a (ax + b ≤ c)
2. Introduceți Constant b
3. Introduceți Right-hand side c
4. Faceți clic pe butonul Calculați
5. Citiți rezultatul afișat sub calculator

Solving Linear Inequalities

A linear inequality is similar to a linear equation but uses inequality signs (>, <, ≥, ≤) instead of =. Our calculator solves inequalities of the form ax + b ≤ c (or any inequality direction). Solving steps are identical to solving equations, with one critical exception: multiplying or dividing by a negative number reverses the inequality direction.

Example: solve 2x + 3 ≤ 11. Subtract 3 from both sides: 2x ≤ 8. Divide by 2 (positive, direction unchanged): x ≤ 4. Solution set: (-∞, 4]. Example with sign flip: solve -3x + 1 > 7. Subtract 1: -3x > 6. Divide by -3 (negative, flip!): x < -2. Solution set: (-∞, -2).

Solution notation: square brackets [ ] indicate the endpoint is included (≤ or ≥), parentheses ( ) indicate it's excluded (< or >). Infinity always uses parentheses. The solution set x ≤ 4 is written (-∞, 4]; x > -2 is written (-2, +∞); -3 < x ≤ 5 is written (-3, 5].

Compound Inequalities and Absolute Value Inequalities

Compound inequalities combine two inequalities. 'And' (conjunction): -2 < x ≤ 5 means x > -2 AND x ≤ 5. Solution is the intersection of both sets. Written as (-2, 5]. 'Or' (disjunction): x < -1 or x ≥ 3. Solution is the union: (-∞, -1) ∪ [3, +∞). 'And' inequalities create bounded intervals; 'or' inequalities create unbounded solutions.

Absolute value inequalities convert to compound inequalities. |x - 3| < 5 means -5 < x - 3 < 5, so -2 < x < 8, solution (-2, 8). |x + 1| ≥ 4 means x + 1 ≤ -4 or x + 1 ≥ 4, so x ≤ -5 or x ≥ 3, solution (-∞, -5] ∪ [3, +∞). The key: |A| < b → -b < A < b; |A| > b → A < -b or A > b.

Quadratic inequalities: x² - x - 6 > 0. Factor: (x-3)(x+2) > 0. Roots at x=3 and x=-2 divide the number line into three intervals. Test each interval: x<-2 (positive ✓), -2<x<3 (negative ✗), x>3 (positive ✓). Solution: (-∞,-2) ∪ (3,+∞).

Inequalities in Optimization and Real-World Problems

Inequalities model real-world constraints in optimization problems. Linear programming maximizes or minimizes a linear objective function subject to a system of linear inequalities. The feasible region (all points satisfying all constraints) is a convex polygon; the optimal solution occurs at a vertex. This method underlies airline scheduling, supply chain optimization, and portfolio selection.

In everyday life, inequalities model budget constraints (spending ≤ income), capacity limits (items ≤ shelf space), safety requirements (speed ≤ limit), and quality standards (error rate < 0.001). Engineering design uses inequalities to ensure stress < yield strength, temperature < maximum rating, and weight < structural capacity.

In mathematics, fundamental inequalities include: AM-GM (arithmetic mean ≥ geometric mean), Cauchy-Schwarz, and the Triangle inequality. These appear in mathematical olympiad problems, analysis, and proofs of many other results. Inequalities are often more powerful tools than equations because they describe regions rather than points.

Frequently Asked Questions