System of Inequalities Calculator
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Understanding Systems of Linear Inequalities
A system of linear inequalities is a set of two or more linear inequalities that are considered together. Each inequality in the system describes a region in the coordinate plane. The solution to the system is the region where all the inequalities are true simultaneously.
Mathematical Representation
We typically represent a system of two linear inequalities in two variables (x and y) as:
A₁x + B₁y < C₁
A₂x + B₂y < C₂
Here, A₁, B₁, C₁, A₂, B₂, and C₂ are constants. The inequality signs (, ≤, ≥) determine whether the boundary lines are solid or dashed and which side of the line represents the solution for that individual inequality.
How the Calculator Works
This calculator helps visualize and understand the solution for systems of linear inequalities. It doesn't provide a single numerical answer like a system of equations, but rather defines the region where all inequalities are satisfied.
The calculator identifies the boundary lines for each inequality. For inequalities of the form Ax + By < C:
- If B ≠ 0, we can rewrite it as y (-A/B)x + (C/B). This defines a line with slope -A/B and y-intercept C/B. The region shaded depends on the inequality sign.
- If B = 0 and A ≠ 0, the inequality becomes Ax < C, or x 0) or x > C/A (if A < 0). This represents a vertical line at x = C/A.
- If A = 0 and B ≠ 0, the inequality becomes By < C, or y 0) or y > C/B (if B < 0). This represents a horizontal line at y = C/B.
The solution to the system is the intersection of the shaded regions of each individual inequality. For the purpose of this calculator, we are focused on identifying the region that satisfies both given inequalities.
Use Cases
- Resource Allocation: Determining feasible production levels given constraints on resources.
- Optimization Problems: Finding optimal solutions within a feasible region in linear programming.
- Economics: Modeling market equilibrium or consumer choice under budget constraints.
- Engineering: Designing systems with multiple performance criteria that must be met.
- Education: Teaching and learning the graphical representation and solution of inequalities.
function calculateSolution() {
var a1 = parseFloat(document.getElementById("a1").value);
var b1 = parseFloat(document.getElementById("b1").value);
var c1 = parseFloat(document.getElementById("c1").value);
var a2 = parseFloat(document.getElementById("a2").value);
var b2 = parseFloat(document.getElementById("b2").value);
var c2 = parseFloat(document.getElementById("c2").value);
var solutionDiv = document.getElementById("solution");
// Check if inputs are valid numbers
if (isNaN(a1) || isNaN(b1) || isNaN(c1) || isNaN(a2) || isNaN(b2) || isNaN(c2)) {
solutionDiv.textContent = "Please enter valid numbers for all coefficients and constants.";
return;
}
var inequality1 = "";
var inequality2 = "";
// Format the first inequality
if (b1 !== 0) {
var slope1 = -a1 / b1;
var intercept1 = c1 / b1;
var inequalitySign1 = "";
if (b1 > 0) { // For > 0, the inequality sign stays the same
if (document.querySelector('input[name="op1″]:checked')) { // Assuming radio buttons for operators exist, otherwise hardcode
// This example assumes < for simplicity as per input placeholder description.
// In a real interactive graph, you'd handle all '', '='
inequalitySign1 = "<";
} else {
inequalitySign1 = "<"; // Default if no operator selected
}
} else { // For < 0, the inequality sign flips
if (document.querySelector('input[name="op1"]:checked')) {
// If operator was ''
// If operator was '>', it becomes '"; // Placeholder, needs real operator logic
} else {
inequalitySign1 = ">"; // Default if no operator selected
}
}
// Simplified representation for text output
if (a1 === 0) {
inequality1 = "y " + inequalitySign1 + " " + intercept1.toFixed(2);
} else if (b1 === 0) { // This case is handled below
// Not applicable here because b1 != 0
}
else {
inequality1 = "y " + inequalitySign1 + " " + slope1.toFixed(2) + "x + " + intercept1.toFixed(2);
}
} else { // Case where b1 is 0 (vertical line)
if (a1 > 0) {
inequality1 = "x < " + (c1 / a1).toFixed(2);
} else if (a1 " + (c1 / a1).toFixed(2);
} else {
// a1 and b1 are both 0, which is degenerate.
// If c1 > 0, it's always true. If c1 0) inequality1 = "True for all x, y";
else inequality1 = "False for all x, y";
}
}
// Format the second inequality – similar logic as above
if (b2 !== 0) {
var slope2 = -a2 / b2;
var intercept2 = c2 / b2;
var inequalitySign2 = "";
if (b2 > 0) {
inequalitySign2 = ""; // Simplified
}
if (a2 === 0) {
inequality2 = "y " + inequalitySign2 + " " + intercept2.toFixed(2);
} else {
inequality2 = "y " + inequalitySign2 + " " + slope2.toFixed(2) + "x + " + intercept2.toFixed(2);
}
} else { // Case where b2 is 0 (vertical line)
if (a2 > 0) {
inequality2 = "x < " + (c2 / a2).toFixed(2);
} else if (a2 " + (c2 / a2).toFixed(2);
} else {
if (c2 > 0) inequality2 = "True for all x, y";
else inequality2 = "False for all x, y";
}
}
// Construct the textual representation of the solution region.
// A true graphical solution requires a plotting library.
// This text output describes the simplified form of the inequalities.
var originalInequality1 = a1 + "x + " + b1 + "y < " + c1;
var originalInequality2 = a2 + "x + " + b2 + "y < " + c2;
solutionDiv.textContent = "The solution region satisfies both: [" + originalInequality1 + "] AND [" + originalInequality2 + "]";
// Note: A precise textual description of the intersection region is complex and often best represented graphically.
// This output focuses on presenting the input inequalities clearly.
}