An absolute value equation involves the absolute value of an expression. The absolute value of a number is its distance from zero on the number line, and it's always non-negative. For an expression like |x|, it means x if x is positive or zero, and -x if x is negative. This property leads to two possibilities when solving equations containing absolute values.
The Form: |ax + b| = c
The most common form of absolute value equations we encounter in algebra is |ax + b| = c, where:
a and b are coefficients/constants within the absolute value.
c is a non-negative constant on the right-hand side of the equation.
Solving Absolute Value Equations
To solve an equation of the form |ax + b| = c, we must consider two cases based on the definition of absolute value:
Case 1: The expression inside the absolute value is non-negative.
If ax + b is non-negative, then |ax + b| = ax + b. The equation becomes:
ax + b = c
Solving for x gives: ax = c - b, so x = (c - b) / a.
Case 2: The expression inside the absolute value is negative.
If ax + b is negative, then |ax + b| = -(ax + b). The equation becomes:
-(ax + b) = c
This is equivalent to ax + b = -c.
Solving for x gives: ax = -c - b, so x = (-c - b) / a.
Important Considerations:
The value of 'c': If c is negative, there are no real solutions because the absolute value of any expression cannot be negative.
The value of 'a': If a is zero, the equation simplifies. If a=0 and b=c, there are infinitely many solutions (all real numbers). If a=0 and b!=c, there are no solutions. Our calculator assumes a is non-zero for simplicity in the division step.
Example Usage:
Let's solve the equation |2x + 4| = 6.
Here, a = 2, b = 4, and c = 6.
Case 1:2x + 4 = 6 => 2x = 2 => x = 1.
Case 2:2x + 4 = -6 => 2x = -10 => x = -5.
The solutions are x = 1 and x = -5.
function calculateAbsoluteValueSolutions() {
var a = parseFloat(document.getElementById("coefficientA").value);
var b = parseFloat(document.getElementById("constantB").value);
var c = parseFloat(document.getElementById("constantC").value);
var resultDiv = document.getElementById("result");
resultDiv.innerHTML = "Solutions will appear here."; // Reset
if (isNaN(a) || isNaN(b) || isNaN(c)) {
resultDiv.innerHTML = "Please enter valid numbers for all fields.";
return;
}
if (c < 0) {
resultDiv.innerHTML = "No real solutions (since c is negative).";
return;
}
if (a === 0) {
if (b === c || b === -c) {
resultDiv.innerHTML = "Infinite solutions (since a=0 and |b|=c).";
} else {
resultDiv.innerHTML = "No solutions (since a=0 and |b|!=c).";
}
return;
}
var solution1 = (c – b) / a;
var solution2 = (-c – b) / a;
// Format solutions to avoid unnecessary trailing zeros if they are integers
var formattedSolution1 = Number.isInteger(solution1) ? solution1 : solution1.toFixed(4);
var formattedSolution2 = Number.isInteger(solution2) ? solution2 : solution2.toFixed(4);
// Remove trailing .0 for integers if they exist after toFixed
formattedSolution1 = formattedSolution1.replace(/\.0$/, '');
formattedSolution2 = formattedSolution2.replace(/\.0$/, '');
if (solution1 === solution2) {
resultDiv.innerHTML = "Unique Solution: x = " + formattedSolution1 + "";
} else {
resultDiv.innerHTML = "Solutions: x = " + formattedSolution1 + " and x = " + formattedSolution2 + "";
}
}