Absolute value inequalities are mathematical statements involving the absolute value of an expression and an inequality symbol (<, >, ≤, ≥). The absolute value of a number represents its distance from zero on the number line. Therefore, an absolute value inequality describes all numbers whose distance from zero, or from a specific point, is within a certain range.
The general form of an absolute value inequality can be expressed as:
|ax + b| < c or |ax + b| ≤ c (where c is positive)
|ax + b| > c or |ax + b| ≥ c (where c is positive)
Note: If c is negative, the solution set depends on the inequality type. For |expression| < c or |expression| ≤ c, there is no solution if c is negative. For |expression| > c or |expression| ≥ c, the solution is all real numbers if c is negative.
Solving Absolute Value Inequalities
The method for solving depends on the inequality symbol:
1. For "Less Than" Inequalities (< or ≤):
An inequality of the form |ax + b| < c (or ≤) means the expression ax + b is between -c and c. This can be rewritten as a compound inequality:
-c < ax + b < c (or -c ≤ ax + b ≤ c)
To solve this, you isolate the variable x by performing the same operations on all three parts of the compound inequality.
2. For "Greater Than" Inequalities (> or ≥):
An inequality of the form |ax + b| > c (or ≥) means the expression ax + b is either less than -c OR greater than c. This leads to two separate inequalities:
ax + b < -c OR ax + b > c
(or ax + b ≤ -c OR ax + b ≥ c for ≥)
You solve each of these inequalities independently for x. The solution is the union of the solutions to these two inequalities.
Calculator Usage:
This calculator helps you solve inequalities of the form |Ax + B| Relation C. You need to input:
Expression Value (C): The number on the right side of the inequality (e.g., 5 in |2x + 3| < 5).
Inequality Type: Select the correct symbol (<, ≤, >, ≥).
Variable: The variable used in the expression (usually 'x').
Coefficient (A): The number multiplying the variable (e.g., 2 in |2x + 3| < 5).
Constant (B): The number added to or subtracted from the variable term (e.g., 3 in |2x + 3| < 5).
The calculator will then provide the solution set for the variable.
Example Use Cases:
Absolute value inequalities are used in various fields:
Physics: To define tolerances or margins of error. For example, if a measurement must be within 5 units of a target value, this can be expressed as |measurement – target| ≤ 5.
Engineering: Specifying acceptable ranges for material properties or component dimensions.
Computer Science: In algorithms that require values to fall within a certain range or distance.
Statistics: Defining confidence intervals or bounds for data points.
function calculateInequality() {
var expressionValue = parseFloat(document.getElementById("expressionValue").value);
var inequalityType = document.getElementById("inequalityType").value;
var variable = document.getElementById("variable").value.trim();
var coefficient = parseFloat(document.getElementById("coefficient").value);
var constant = parseFloat(document.getElementById("constant").value);
var resultDiv = document.getElementById("result");
resultDiv.innerHTML = ""; // Clear previous results
// Input validation
if (isNaN(expressionValue) || isNaN(coefficient) || isNaN(constant)) {
resultDiv.innerHTML = "Error: Please enter valid numbers for all fields.";
return;
}
if (variable === "") {
resultDiv.innerHTML = "Error: Please enter a variable.";
return;
}
if (expressionValue < 0 && (inequalityType === "less_than" || inequalityType === "less_than_equal")) {
resultDiv.innerHTML = "Solution: No solution (absolute value cannot be less than a negative number).";
return;
}
if (expressionValue ",
"greater_than_equal": ">=",
"less_than": "<",
"less_than_equal": "<="
};
inequalitySymbolDisplay = inequalityTypeMap[inequalityType];
if (inequalityType === "less_than" || inequalityType === "less_than_equal") {
// |Ax + B| -C < Ax + B < C
// |Ax + B| -C <= Ax + B =";
var inequality1Op2 = "";
inequality1Op2 = " 0) {
// (lowerBound – constant) / coefficient < x < (upperBound – constant) / coefficient
var startX = (lowerBound – constant) / coefficient;
var endX = (upperBound – constant) / coefficient;
solution = variable + " " + inequality1Op1 + " " + formatNumber(startX) + " " + inequality1Op2 + " " + variable + " " + formatNumber(endX);
} else if (coefficient < 0) {
// When dividing by a negative coefficient, flip the inequality signs
// (upperBound – constant) / coefficient < x = lowerBound && constant C => Ax + B C
// |Ax + B| >= C => Ax + B = C
var negativeExpressionValue = -expressionValue;
var inequality1Op = "";
if (inequalityType === "greater_than_equal") {
inequality1Op = "=";
}
var part1Solution = "";
var part2Solution = "";
// First part: Ax + B relation negativeExpressionValue
if (coefficient > 0) {
// x relation (negativeExpressionValue – constant) / coefficient
var val1 = (negativeExpressionValue – constant) / coefficient;
part1Solution = variable + " " + inequality1Op + " " + formatNumber(val1);
} else if (coefficient < 0) {
// x relation (negativeExpressionValue – constant) / coefficient (sign flipped)
var val1 = (negativeExpressionValue – constant) / coefficient;
if (inequality1Op === " " + formatNumber(val1);
else if (inequality1Op === "= " + formatNumber(val1);
else if (inequality1Op === ">") part1Solution = variable + " =") part1Solution = variable + " <= " + formatNumber(val1);
} else { // coefficient is 0
if (constant < negativeExpressionValue && inequality1Op === "<") part1Solution = "All real numbers";
else if (constant <= negativeExpressionValue && inequality1Op === " 0) {
// x relation (expressionValue – constant) / coefficient
var val2 = (expressionValue – constant) / coefficient;
part2Solution = variable + " " + inequality2Op + " " + formatNumber(val2);
} else if (coefficient < 0) {
// x relation (expressionValue – constant) / coefficient (sign flipped)
var val2 = (expressionValue – constant) / coefficient;
if (inequality2Op === " " + formatNumber(val2);
else if (inequality2Op === "= " + formatNumber(val2);
else if (inequality2Op === ">") part2Solution = variable + " =") part2Solution = variable + " expressionValue && inequality2Op === ">") part2Solution = "All real numbers";
else if (constant >= expressionValue && inequality2Op === ">=") part2Solution = "All real numbers";
else part2Solution = "No solution";
}
if (part1Solution === "All real numbers" || part2Solution === "All real numbers") {
solution = "All real numbers";
} else if (part1Solution === "No solution" && part2Solution === "No solution") {
solution = "No solution";
}
else {
solution = part1Solution + " OR " + part2Solution;
}
}
if (solution) {
resultDiv.innerHTML = "Solution: " + solution;
} else {
resultDiv.innerHTML = "Could not determine solution. Check inputs.";
}
}
// Helper function to format numbers nicely, avoiding excessive decimals
function formatNumber(num) {
if (num === Infinity) return "∞";
if (num === -Infinity) return "-∞";
// Check if the number is an integer
if (num === Math.round(num)) {
return num.toString();
}
// Otherwise, format to a reasonable number of decimal places
return num.toFixed(4).replace(/\.?0+$/, "); // Remove trailing zeros and decimal point if applicable
}