This calculator helps solve equations of the form |ax + b| = c.
Results will appear here.
Understanding Absolute Value Equations
An absolute value equation involves the absolute value of an expression, typically in the form |expression| = number.
The absolute value of a number is its distance from zero on the number line, and it's always non-negative. For example, |5| = 5 and |-5| = 5.
A common type of absolute value equation is of the form |ax + b| = c. To solve this, we must consider two cases because the expression inside the absolute value bars (ax + b) could be either positive or negative:
Case 1: The expression inside is positive (or zero)
If ax + b is positive or zero, then ax + b = c.
To solve for x in this case:
Subtract 'b' from both sides: ax = c - b
Divide by 'a': x = (c - b) / a
Case 2: The expression inside is negative
If ax + b is negative, then its absolute value is -(ax + b). So, -(ax + b) = c.
To solve for x in this case:
Distribute the negative sign: -ax - b = c
Add 'b' to both sides: -ax = c + b
Divide by '-a': x = (c + b) / -a, which is equivalent to x = -(c + b) / a
Important Considerations:
Non-Negative Right Side: If the value on the right side of the equation (c) is negative, there are no real solutions because the absolute value of any expression is always non-negative.
Coefficient 'a': If the coefficient 'a' is zero, the equation simplifies. If a=0, the equation becomes |b| = c. This equation has solutions only if |b| == c. If b=c or b=-c, any real number 'x' is a solution (if a=0 and b=0 and c=0). If |b| != c, there are no solutions. Our calculator handles the case where a is not zero.
Example: Solve |2x + 3| = 7
Here, a = 2, b = 3, and c = 7.
Since c (7) is positive, we expect solutions.
Case 1:
2x + 3 = 7
2x = 7 - 3
2x = 4
x = 4 / 2
x = 2
Case 2:
-(2x + 3) = 7
-2x - 3 = 7
-2x = 7 + 3
-2x = 10
x = 10 / -2
x = -5
The solutions are x = 2 and x = -5.
function solveAbsoluteValueEquation() {
var a = parseFloat(document.getElementById("coefficientA").value);
var b = parseFloat(document.getElementById("constantB").value);
var c = parseFloat(document.getElementById("rightSideC").value);
var resultDiv = document.getElementById("result");
resultDiv.innerHTML = 'Results will appear here.'; // Clear previous results
// Input validation
if (isNaN(a) || isNaN(b) || isNaN(c)) {
resultDiv.innerHTML = 'Please enter valid numbers for all fields.';
return;
}
// Handle the case where the right side (c) is negative
if (c < 0) {
resultDiv.innerHTML = 'No real solutions exist because the absolute value cannot be negative.';
return;
}
// Handle the case where coefficient 'a' is zero
if (a === 0) {
if (Math.abs(b) === c) {
resultDiv.innerHTML = 'The equation is always true for any real number x.';
} else {
resultDiv.innerHTML = 'No solution exists for this equation.';
}
return;
}
// Calculate the two possible solutions
var x1 = (c – b) / a;
var x2 = -(c + b) / a;
// Display the results
resultDiv.innerHTML = 'Solution 1: x = ' + x1.toFixed(4) + '' +
'Solution 2: x = ' + x2.toFixed(4) + '';
// Optional: Add a check if the two solutions are the same (e.g., when c = 0)
if (Math.abs(x1 – x2) < 1e-9) { // Using a small tolerance for float comparison
resultDiv.innerHTML = 'The single solution is: x = ' + x1.toFixed(4) + '';
}
}