Inequality Calculator with Steps
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Inequality Calculator with Steps
Solution
Enter values to see the solution.
Calculation Steps:
Enter valid inputs above to see the step-by-step solution.
Understanding and Solving Quadratic Inequalities
Quadratic inequalities are mathematical statements involving a quadratic expression and an inequality sign. They are used to define regions or intervals where a quadratic function satisfies a certain condition. The general form of a quadratic inequality is:
ax² + bx + c > 0ax² + bx + c < 0ax² + bx + c ≥ 0ax² + bx + c ≤ 0
where a, b, and c are coefficients, and a ≠ 0. If a = 0, the inequality becomes linear.
Why Use an Inequality Calculator?
Solving quadratic inequalities by hand can be a meticulous process involving finding roots, testing intervals, and considering the parabola's shape. An inequality calculator automates these steps, providing quick and accurate solutions. This is particularly useful for:
- Students: Verifying their manual calculations and understanding the process.
- Engineers and Scientists: Quickly determining operational ranges or conditions where a system behaves within certain bounds.
- Data Analysts: Identifying data ranges that meet specific criteria.
How the Calculator Works:
This calculator solves quadratic inequalities of the form ax² + bx + c [inequality sign] 0. Here's a breakdown of the mathematical steps involved:
Step 1: Find the Roots
The first step is to find the roots of the corresponding quadratic equation ax² + bx + c = 0. These roots are the points where the quadratic function f(x) = ax² + bx + c equals zero. They divide the number line into intervals.
The roots are found using the quadratic formula:
x = [-b ± sqrt(b² - 4ac)] / 2a
The term Δ = b² - 4ac is called the discriminant.
- If
Δ > 0, there are two distinct real roots.
- If
Δ = 0, there is exactly one real root (a repeated root).
- If
Δ < 0, there are no real roots.
Step 2: Determine the Intervals
The real roots found in Step 1 divide the number line into sections. For example, if the roots are r1 and r2 (with r1 < r2), the number line is divided into three intervals: (-∞, r1), (r1, r2), and (r2, ∞).
Step 3: Test Intervals (or Analyze Parabola Shape)
There are two primary methods to determine which intervals satisfy the inequality:
- Test Points: Choose a test value within each interval and substitute it into the original inequality
ax² + bx + c [inequality sign] 0. If the inequality holds true for the test value, the entire interval is part of the solution.
- Parabola Shape: Analyze the shape of the parabola represented by
y = ax² + bx + c.
- If
a > 0, the parabola opens upwards. The function is positive outside the roots and negative between the roots.
- If
a < 0, the parabola opens downwards. The function is negative outside the roots and positive between the roots.
The calculator uses the parabola's shape for efficiency when real roots exist. If there are no real roots, the inequality is either true for all real numbers (if the parabola is entirely above or below the x-axis) or false for all real numbers.
Step 4: Consider the Inequality Sign
The type of inequality sign (>, <, ≥, ≤) determines whether the endpoints (the roots) are included in the solution set.
- For
> and <, the roots are not included (open intervals).
- For
≥ and ≤, the roots are included (closed intervals).
- For
== and !=, the solution consists only of the roots themselves or all real numbers except the roots, respectively.
Example Calculation: x² - 5x + 6 > 0
Let's use the calculator's logic for this example:
- Coefficients:
a = 1, b = -5, c = 6
- Inequality Type:
>
- Step 1: Find Roots
- Solve
x² - 5x + 6 = 0.
- Discriminant:
Δ = (-5)² - 4(1)(6) = 25 - 24 = 1.
- Roots:
x = [ -(-5) ± sqrt(1) ] / (2 * 1) = [ 5 ± 1 ] / 2.
x1 = (5 - 1) / 2 = 4 / 2 = 2x2 = (5 + 1) / 2 = 6 / 2 = 3- The roots are 2 and 3.
- Step 2: Determine Intervals
- The roots 2 and 3 divide the number line into:
(-∞, 2), (2, 3), and (3, ∞).
- Step 3 & 4: Analyze Parabola & Inequality
- Since
a = 1 > 0, the parabola opens upwards.
- The inequality is
> 0 (greater than zero), meaning we are looking for where the parabola is above the x-axis.
- For an upward-opening parabola, the function is positive (above the x-axis) outside the roots.
- The inequality sign is
>, so the roots are not included.
- Solution: The inequality is true for
x < 2 or x > 3. In interval notation: (-∞, 2) U (3, ∞).
function calculateInequality() {
var a = parseFloat(document.getElementById("coefficientA").value);
var b = parseFloat(document.getElementById("coefficientB").value);
var c = parseFloat(document.getElementById("constantC").value);
var inequalityType = document.getElementById("inequalityType").value;
var resultDiv = document.getElementById("result");
var stepsDiv = document.getElementById("steps");
var stepsHtml = "
Calculation Steps:
";
// Input validation
if (isNaN(a) || isNaN(b) || isNaN(c)) {
resultDiv.innerHTML = "Error: Please enter valid numbers for all coefficients.";
stepsDiv.innerHTML = stepsHtml + "Invalid input detected. Please ensure all coefficient fields contain valid numbers.";
return;
}
// Handle linear case if a is 0
if (a === 0) {
stepsHtml += "Coefficient 'a' is 0. This is a linear inequality, not quadratic.";
if (b === 0) {
if (inequalityType === ">") {
if (c > 0) {
resultDiv.innerHTML = "All Real Numbers";
stepsHtml += "0x + 0 + c > 0 => c > 0. Since c is positive, the inequality is true for all real numbers.";
} else {
resultDiv.innerHTML = "No Solution";
stepsHtml += "0x + 0 + c > 0 => c > 0. Since c is not positive, there is no solution.";
}
} else if (inequalityType === "<") {
if (c < 0) {
resultDiv.innerHTML = "All Real Numbers";
stepsHtml += "0x + 0 + c c < 0. Since c is negative, the inequality is true for all real numbers.";
} else {
resultDiv.innerHTML = "No Solution";
stepsHtml += "0x + 0 + c c =") {
if (c >= 0) {
resultDiv.innerHTML = "All Real Numbers";
stepsHtml += "0x + 0 + c >= 0 => c >= 0. Since c is non-negative, the inequality is true for all real numbers.";
} else {
resultDiv.innerHTML = "No Solution";
stepsHtml += "0x + 0 + c >= 0 => c >= 0. Since c is negative, there is no solution.";
}
} else if (inequalityType === "<=") {
if (c <= 0) {
resultDiv.innerHTML = "All Real Numbers";
stepsHtml += "0x + 0 + c c <= 0. Since c is non-positive, the inequality is true for all real numbers.";
} else {
resultDiv.innerHTML = "No Solution";
stepsHtml += "0x + 0 + c c c == 0. Since c is 0, the equality holds for all real numbers.";
} else {
resultDiv.innerHTML = "No Solution";
stepsHtml += "0x + 0 + c == 0 => c == 0. Since c is not 0, there is no solution.";
}
} else if (inequalityType === "!=") {
if (c !== 0) {
resultDiv.innerHTML = "All Real Numbers";
stepsHtml += "0x + 0 + c != 0 => c != 0. Since c is not 0, the inequality holds for all real numbers.";
} else {
resultDiv.innerHTML = "No Solution";
stepsHtml += "0x + 0 + c != 0 => c != 0. Since c is 0, there is no solution.";
}
}
} else { // Linear case: bx + c [inequality] 0
var x;
var inequalitySign = inequalityType;
var effectiveInequality = inequalitySign;
if (b ") effectiveInequality = "<";
else if (inequalitySign === "";
else if (inequalitySign === ">=") effectiveInequality = "<=";
else if (inequalitySign === "=";
}
var solutionSet = "";
if (inequalityType === ">") {
if (b > 0) { x = -c / b; solutionSet = "x > " + x; }
else { x = -c / b; solutionSet = "x < " + x; }
} else if (inequalityType === " 0) { x = -c / b; solutionSet = "x " + x; }
} else if (inequalityType === ">=") {
if (b > 0) { x = -c / b; solutionSet = "x >= " + x; }
else { x = -c / b; solutionSet = "x <= " + x; }
} else if (inequalityType === " 0) { x = -c / b; solutionSet = "x = " + x; }
} else if (inequalityType === "==") {
x = -c / b; solutionSet = "x = " + x;
} else if (inequalityType === "!=") {
x = -c / b; solutionSet = "x != " + x;
}
resultDiv.innerHTML = effectiveInequality.replace(">", ">").replace("<", " 0 ? "- " + Math.abs(c) : "+ " + Math.abs(c)) + " 0″;
stepsHtml += "Move the constant term: " + b + "x " + (c > 0 ? "- " + Math.abs(c) : "+ " + Math.abs(c)) + " ";
if (b < 0) {
stepsHtml += "Divide by " + b + " (negative). Remember to flip the inequality sign.";
} else {
stepsHtml += "Divide by " + b + ".";
}
stepsHtml += "Solution: " + solutionSet.replace("<", "", ">") + "";
}
stepsDiv.innerHTML = stepsHtml;
return;
}
stepsHtml += "We need to solve the inequality: " + a + "x² + " + b + "x + " + c + " " + inequalityType + " 0″;
// Step 1: Find the roots of the quadratic equation ax² + bx + c = 0
var discriminant = b * b – 4 * a * c;
stepsHtml += "
Step 1: Find the roots of the corresponding equation " + a + "x² + " + b + "x + " + c + " = 0";
stepsHtml += "Calculate the discriminant (Δ): Δ = b² – 4ac";
stepsHtml += "Δ = (" + b + ")² – 4 * (" + a + ") * (" + c + ") = " + b * b + " – " + (4 * a * c) + " = " + discriminant + "";
var roots = [];
if (discriminant >= 0) {
var sqrtDiscriminant = Math.sqrt(discriminant);
var root1 = (-b – sqrtDiscriminant) / (2 * a);
var root2 = (-b + sqrtDiscriminant) / (2 * a);
roots.push(root1, root2);
stepsHtml += "Since Δ ≥ 0, there are real roots.";
stepsHtml += "Using the quadratic formula x = [-b ± sqrt(Δ)] / 2a:";
stepsHtml += "Root 1: x₁ = [ -(" + b + ") – sqrt(" + discriminant + ") ] / (2 * " + a + ") = [ " + (-b) + " – " + sqrtDiscriminant + " ] / " + (2 * a) + " = " + root1 + "";
stepsHtml += "Root 2: x₂ = [ -(" + b + ") + sqrt(" + discriminant + ") ] / (2 * " + a + ") = [ " + (-b) + " + " + sqrtDiscriminant + " ] / " + (2 * a) + " = " + root2 + "";
if (root1 === root2) {
stepsHtml += "There is one repeated real root: x = " + root1 + "";
roots = [root1]; // Ensure roots array has only one element for single root case
} else {
// Sort roots for consistent interval definition
roots.sort(function(x, y) { return x – y; });
stepsHtml += "The distinct real roots are " + roots[0] + " and " + roots[1] + ".";
}
} else {
stepsHtml += "Since Δ < 0, there are no real roots for the equation.";
}
// Step 2 & 3: Determine intervals and analyze parabola
stepsHtml += "
Step 2 & 3: Analyze the parabola and determine solution intervals";
var solution = "";
var solutionIntervals = [];
if (discriminant 0) { // Parabola opens upwards, always above x-axis
if (inequalityType === ">" || inequalityType === ">=") {
solution = "All Real Numbers";
stepsHtml += "The parabola opens upwards (a > 0) and has no real roots, meaning it is always above the x-axis (always positive).";
stepsHtml += "For the inequality " + inequalityType + " 0, the solution is all real numbers.";
} else { // "<" or " 0) and has no real roots, meaning it is always above the x-axis (always positive).";
stepsHtml += "For the inequality " + inequalityType + " 0, there is no solution.";
}
} else { // a < 0: Parabola opens downwards, always below x-axis
if (inequalityType === "<" || inequalityType === "<=") {
solution = "All Real Numbers";
stepsHtml += "The parabola opens downwards (a " or ">="
solution = "No Solution";
stepsHtml += "The parabola opens downwards (a 1) ? roots[1] : r1; // Handle single root case
stepsHtml += "The roots " + r1 + (roots.length > 1 ? " and " + r2 : "") + " divide the number line into intervals.";
if (a > 0) { // Parabola opens upwards
stepsHtml += "Since a (" + a + ") is positive, the parabola opens upwards. The function is positive outside the roots and negative between the roots.";
if (inequalityType === ">") {
solution = "x " + r2;
solutionIntervals.push("(-∞, " + r1 + ")");
if (roots.length > 1) solutionIntervals.push("(" + r2 + ", ∞)");
stepsHtml += "We are looking for where the function is greater than 0 (positive). This occurs outside the roots.";
} else if (inequalityType === " " + r1 + " and x 1) solutionIntervals.push("(" + r1 + ", " + r2 + ")");
stepsHtml += "We are looking for where the function is less than 0 (negative). This occurs between the roots.";
} else if (inequalityType === ">=") {
solution = "x = " + r2;
solutionIntervals.push("(-∞, " + r1 + "]");
if (roots.length > 1) solutionIntervals.push("[" + r2 + ", ∞)");
stepsHtml += "We are looking for where the function is greater than or equal to 0. This occurs at the roots and outside the roots.";
} else if (inequalityType === "= " + r1 + " and x 1) solutionIntervals.push("[" + r1 + ", " + r2 + "]");
stepsHtml += "We are looking for where the function is less than or equal to 0. This occurs at the roots and between the roots.";
} else if (inequalityType === "==") {
solution = "x = " + r1 + (roots.length > 1 ? " or x = " + r2 : "");
stepsHtml += "We are looking for where the function is exactly 0. This occurs only at the roots.";
} else if (inequalityType === "!=") {
solution = "x != " + r1 + (roots.length > 1 ? " and x != " + r2 : "");
stepsHtml += "We are looking for where the function is not 0. This occurs everywhere except at the roots.";
if (roots.length > 1) {
solutionIntervals.push("(-∞, " + r1 + ")");
solutionIntervals.push("(" + r1 + ", " + r2 + ")");
solutionIntervals.push("(" + r2 + ", ∞)");
} else {
solutionIntervals.push("(-∞, " + r1 + ")");
solutionIntervals.push("(" + r1 + ", ∞)");
}
}
} else { // a ") {
solution = "x > " + r1 + " and x 1) solutionIntervals.push("(" + r1 + ", " + r2 + ")");
stepsHtml += "We are looking for where the function is greater than 0 (positive). This occurs between the roots.";
} else if (inequalityType === "<") {
solution = "x " + r2;
solutionIntervals.push("(-∞, " + r1 + ")");
if (roots.length > 1) solutionIntervals.push("(" + r2 + ", ∞)");
stepsHtml += "We are looking for where the function is less than 0 (negative). This occurs outside the roots.";
} else if (inequalityType === ">=") {
solution = "x >= " + r1 + " and x 1) solutionIntervals.push("[" + r1 + ", " + r2 + "]");
stepsHtml += "We are looking for where the function is greater than or equal to 0. This occurs at the roots and between the roots.";
} else if (inequalityType === "<=") {
solution = "x = " + r2;
solutionIntervals.push("(-∞, " + r1 + "]");
if (roots.length > 1) solutionIntervals.push("[" + r2 + ", ∞)");
stepsHtml += "We are looking for where the function is less than or equal to 0. This occurs at the roots and outside the roots.";
} else if (inequalityType === "==") {
solution = "x = " + r1 + (roots.length > 1 ? " or x = " + r2 : "");
stepsHtml += "We are looking for where the function is exactly 0. This occurs only at the roots.";
} else if (inequalityType === "!=") {
solution = "x != " + r1 + (roots.length > 1 ? " and x != " + r2 : "");
stepsHtml += "We are looking for where the function is not 0. This occurs everywhere except at the roots.";
if (roots.length > 1) {
solutionIntervals.push("(-∞, " + r1 + ")");
solutionIntervals.push("(" + r1 + ", " + r2 + ")");
solutionIntervals.push("(" + r2 + ", ∞)");
} else {
solutionIntervals.push("(-∞, " + r1 + ")");
solutionIntervals.push("(" + r1 + ", ∞)");
}
}
}
}
// Format the final result and steps
var formattedSolution = solution.replace("<", "", ">");
resultDiv.innerHTML = formattedSolution;
if (solutionIntervals.length > 0 && solution !== "All Real Numbers" && solution !== "No Solution") {
stepsHtml += "
Step 4: Combine intervals based on the inequality sign";
stepsHtml += "The solution set is: " + solutionIntervals.join(" U ") + "";
} else if (solution === "All Real Numbers") {
stepsHtml += "The solution set is all real numbers.";
} else if (solution === "No Solution") {
stepsHtml += "There is no value of x that satisfies this inequality.";
}
stepsDiv.innerHTML = stepsHtml;
}