Modulus Equation Calculator

Modulus Equation Solver: |Ax + B| = C

function calculateModulusEquation() { var A = parseFloat(document.getElementById("coeffA").value); var B = parseFloat(document.getElementById("constantB").value); var C = parseFloat(document.getElementById("constantC").value); var resultDiv = document.getElementById("modulusResult"); resultDiv.innerHTML = ""; // Clear previous results if (isNaN(A) || isNaN(B) || isNaN(C)) { resultDiv.innerHTML = "Please enter valid numbers for all fields."; return; } if (A === 0) { // Special case: |B| = C if (Math.abs(B) === C) { resultDiv.innerHTML = "The equation is true for all real numbers x."; } else { resultDiv.innerHTML = "No real solutions for x."; } return; } if (C < 0) { resultDiv.innerHTML = "No real solutions for x (since the absolute value cannot be negative)."; } else if (C === 0) { // Ax + B = 0 var x = -B / A; resultDiv.innerHTML = "One solution: x = " + x.toFixed(4) + ""; } else { // Ax + B = C OR Ax + B = -C var x1 = (C – B) / A; var x2 = (-C – B) / A; if (x1.toFixed(4) === x2.toFixed(4)) { // Compare fixed values to handle potential floating point inaccuracies for very close numbers resultDiv.innerHTML = "Solution: x = " + x1.toFixed(4) + ""; } else { resultDiv.innerHTML = "Two solutions:"; resultDiv.innerHTML += "x1 = " + x1.toFixed(4) + ""; resultDiv.innerHTML += "x2 = " + x2.toFixed(4) + ""; } } } .calculator-container { font-family: Arial, sans-serif; background-color: #f9f9f9; border: 1px solid #ddd; padding: 20px; border-radius: 8px; max-width: 500px; margin: 20px auto; box-shadow: 0 2px 4px rgba(0,0,0,0.1); } .calculator-container h2 { text-align: center; color: #333; margin-bottom: 20px; } .input-group { margin-bottom: 15px; } .input-group label { display: block; margin-bottom: 5px; font-weight: bold; color: #555; } .input-group input[type="number"] { width: calc(100% – 22px); padding: 10px; border: 1px solid #ccc; border-radius: 4px; box-sizing: border-box; font-size: 16px; } button { background-color: #007bff; color: white; padding: 12px 20px; border: none; border-radius: 4px; cursor: pointer; font-size: 16px; width: 100%; box-sizing: border-box; transition: background-color 0.3s ease; } button:hover { background-color: #0056b3; } .result-container { margin-top: 20px; padding: 15px; border: 1px solid #e0e0e0; border-radius: 4px; background-color: #eaf6ff; min-height: 50px; color: #333; } .result-container p { margin: 5px 0; font-size: 1.1em; line-height: 1.5; } .result-container .error { color: #d9534f; font-weight: bold; }

Understanding Modulus Equations

A modulus equation, also known as an absolute value equation, is an equation that contains an absolute value expression. The absolute value of a number represents its distance from zero on the number line, regardless of direction. For example, |5| = 5 and |-5| = 5. This means that an absolute value expression can result in two possible values for the expression inside the absolute value bars.

The General Form: |Ax + B| = C

Our calculator focuses on solving modulus equations of the form |Ax + B| = C, where A, B, and C are constants, and x is the variable we want to solve for. Understanding the properties of absolute values is key to solving these equations.

How to Solve Modulus Equations

The core principle for solving |Ax + B| = C depends on the value of C:

Case 1: C is Negative (C < 0)

If C is a negative number, there are no real solutions for x. This is because the absolute value of any real number (or expression) must be non-negative (zero or positive). It cannot equal a negative number.

Example: |2x + 3| = -5
Since the right side is negative, there are no real solutions.

Case 2: C is Zero (C = 0)

If C is zero, then the expression inside the absolute value must be zero. This leads to a single linear equation:

Ax + B = 0

Solving for x gives: x = -B / A (provided A is not zero).

Example: |3x - 6| = 0
3x - 6 = 0
3x = 6
x = 2

Case 3: C is Positive (C > 0)

If C is a positive number, then the expression inside the absolute value can be either C or -C. This leads to two separate linear equations:

  1. Ax + B = C
  2. Ax + B = -C

You solve each of these equations independently to find the two possible values for x.

Example: |2x + 1| = 7

Equation 1:
2x + 1 = 7
2x = 6
x = 3

Equation 2:
2x + 1 = -7
2x = -8
x = -4

So, the solutions are x = 3 and x = -4.

Special Case: When A = 0

If the coefficient A is zero, the equation simplifies to |B| = C.

  • If |B| = C is true (e.g., |5| = 5), then the equation is true for all real numbers x. This means any value of x will satisfy the equation.
  • If |B| = C is false (e.g., |5| = 3), then there are no real solutions for x.

Our calculator handles this special case to provide accurate results.

Using the Calculator

Simply input the values for A, B, and C from your modulus equation |Ax + B| = C into the respective fields. Click "Calculate Solutions" to see the real solutions for x, or a message indicating no solutions or infinite solutions.

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