Enter the coefficients and constants of your linear inequality.
<option value="Less Than (<)
<option value="Less Than or Equal To (≤)
">Greater Than (>)
=">Greater Than or Equal To (≥)
Your solution will appear here.
Understanding and Solving Linear Inequalities
Linear inequalities are mathematical statements that compare two expressions using inequality symbols such as , ≤, or ≥. They are fundamental in mathematics and have wide-ranging applications in optimization, resource allocation, and defining feasible regions in various problems.
The General Form
A simple linear inequality with one variable, say 'x', can often be written in the form:
ax + b < c
or variants with other inequality symbols (≤, >, ≥).
a is the coefficient of the variable 'x'.
b is a constant term added to the variable term.
c is the constant value on the right side of the inequality.
How the Calculator Works
This calculator takes the coefficients and constants of a linear inequality in the form ax + b [symbol] c and provides the solution for 'x'. The process involves algebraic manipulation similar to solving linear equations, with a crucial difference when multiplying or dividing by a negative number.
Steps to Solve:
Isolate the variable term: Subtract 'b' from both sides of the inequality. This gives ax [new b symbol] (c - b).
Isolate 'x': Divide both sides by 'a'.
If 'a' is positive, the inequality symbol remains the same.
If 'a' is negative, the inequality symbol must be reversed (e.g., , > becomes <).
Example Calculation
Let's solve the inequality: 2x + 5 > 15
Here, a = 2, b = 5, inequality symbol is >, and c = 15.
Step 1: Subtract 5 from both sides:
2x > 15 - 52x > 10
Step 2: Divide by 2 (which is positive, so the symbol stays the same):
x > 10 / 2x > 5
The solution is x > 5.
Consider another example: -3x + 1 ≤ 10
Here, a = -3, b = 1, inequality symbol is ≤, and c = 10.
Step 1: Subtract 1 from both sides:
-3x ≤ 10 - 1-3x ≤ 9
Step 2: Divide by -3 (which is negative, so the symbol reverses):
x ≥ 9 / -3x ≥ -3
The solution is x ≥ -3.
Applications
Solving inequalities is crucial in various fields:
Budgeting: Determining how many items can be purchased within a certain budget.
Resource Management: Allocating resources based on constraints.
Engineering: Ensuring performance metrics stay within acceptable ranges.
Computer Science: Analyzing algorithm complexity and defining conditions for execution.
function solveInequality() {
var aInput = document.getElementById("coefficientA");
var bInput = document.getElementById("constantB");
var inequalitySymbol = document.getElementById("inequalitySymbol").value;
var cInput = document.getElementById("rightSideValue");
var resultDiv = document.getElementById("result");
var a = parseFloat(aInput.value);
var b = parseFloat(bInput.value);
var c = parseFloat(cInput.value);
if (isNaN(a) || isNaN(b) || isNaN(c)) {
resultDiv.innerHTML = "Error: Please enter valid numbers for all inputs.";
return;
}
var simplifiedB = c – b;
var solutionX;
var inequalitySignChanged = false;
if (a > 0) {
solutionX = simplifiedB / a;
resultDiv.innerHTML = "Solution: x " + inequalitySymbol + " " + solutionX;
} else if (a < 0) {
solutionX = simplifiedB / a;
var reversedSymbol = inequalitySymbol;
if (inequalitySymbol === "";
} else if (inequalitySymbol === "=";
} else if (inequalitySymbol === ">") {
reversedSymbol = "=") {
reversedSymbol = "<=";
}
resultDiv.innerHTML = "Solution: x " + reversedSymbol + " " + solutionX + " (Inequality symbol reversed)";
inequalitySignChanged = true;
} else { // a === 0
if (inequalitySymbol === "<") {
if (b < c) {
resultDiv.innerHTML = "Solution: True for all real numbers.";
} else {
resultDiv.innerHTML = "Solution: False for all real numbers.";
}
} else if (inequalitySymbol === "<=") {
if (b <= c) {
resultDiv.innerHTML = "Solution: True for all real numbers.";
} else {
resultDiv.innerHTML = "Solution: False for all real numbers.";
}
} else if (inequalitySymbol === ">") {
if (b > c) {
resultDiv.innerHTML = "Solution: True for all real numbers.";
} else {
resultDiv.innerHTML = "Solution: False for all real numbers.";
}
} else if (inequalitySymbol === ">=") {
if (b >= c) {
resultDiv.innerHTML = "Solution: True for all real numbers.";
} else {
resultDiv.innerHTML = "Solution: False for all real numbers.";
}
}
return; // Exit early for a=0 case
}
}