Absolute Value Equation Calculator
Use this calculator to solve absolute value equations of the form |ax + b| = c. Enter the coefficients and constants, and the calculator will provide the real solutions for x, if any exist.
Results:
|ax + b| cannot equal a negative number).';
} else if (a === 0) {
// Special case: |b| = c
if (Math.abs(b) === c) {
output = 'Since a = 0, the equation simplifies to |' + b + '| = ' + c + '. As |' + b + '| equals ' + c + ', this statement is true for all real values of x.';
} else {
output = 'Since a = 0, the equation simplifies to |' + b + '| = ' + c + '. As |' + b + '| does not equal ' + c + ', there are no real solutions.';
}
} else if (c === 0) {
// Case: |ax + b| = 0 implies ax + b = 0
var x = -b / a;
output = 'Since c = 0, the equation simplifies to ' + a + 'x + ' + b + ' = 0.';
output += 'Solution: x = ' + x.toFixed(4) + '';
} else {
// Case: |ax + b| = c implies ax + b = c OR ax + b = -c
var x1 = (c – b) / a;
var x2 = (-c – b) / a;
output = 'The equation |' + a + 'x + ' + b + '| = ' + c + ' leads to two possibilities:';
output += 'Possibility 1: ' + a + 'x + ' + b + ' = ' + c + '';
output += '' + a + 'x = ' + (c - b) + '';
output += 'x = ' + x1.toFixed(4) + '';
output += 'Possibility 2: ' + a + 'x + ' + b + ' = -' + c + '';
output += '' + a + 'x = ' + (-c - b) + '';
output += 'x = ' + x2.toFixed(4) + '';
if (x1.toFixed(4) === x2.toFixed(4)) { // Check if solutions are identical due to floating point or actual identity
output = 'The equation |' + a + 'x + ' + b + '| = ' + c + ' has one unique solution:';
output += 'Solution: x = ' + x1.toFixed(4) + '';
}
}
resultDiv.innerHTML = output;
}
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margin: 20px auto;
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Understanding Absolute Value Equations
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, such as |x|, can never be negative.
The General Form: |ax + b| = c
The most common form of an absolute value equation is |ax + b| = c, where a, b, and c are real numbers, and x is the variable we are solving for.
How to Solve Absolute Value Equations
Case 1: When c < 0 (Negative Constant)
If the constant c on the right side of the equation is negative, there are no real solutions. This is because an absolute value expression (like |ax + b|) can never result in a negative number. For example, |2x + 3| = -5 has no real solutions.
Case 2: When c = 0 (Zero Constant)
If the constant c is zero, the equation simplifies to ax + b = 0. This is because the only number whose absolute value is zero is zero itself. You can then solve this linear equation for x. For example, to solve |x - 4| = 0, you set x - 4 = 0, which gives x = 4.
Case 3: When c > 0 (Positive Constant)
If the constant c is positive, there are typically two possible solutions. This is because the expression inside the absolute value could be equal to c or equal to -c. So, you set up two separate linear equations:
ax + b = cax + b = -c
You then solve each of these equations independently to find the two potential values for x. For example, to solve |3x + 1| = 7:
3x + 1 = 7→3x = 6→x = 23x + 1 = -7→3x = -8→x = -8/3
The solutions are x = 2 and x = -8/3.
Special Case: When a = 0
If the coefficient a is zero, the equation becomes |0x + b| = c, which simplifies to |b| = c. In this scenario:
- If
|b|is indeed equal toc(e.g.,|5| = 5), then the equation is true for all real numbersx. - If
|b|is not equal toc(e.g.,|5| = 3), then there are no real solutions.
Examples Using the Calculator:
- Example 1: Two Solutions
Equation:|2x + 3| = 7
Input:a = 2,b = 3,c = 7
Output:x = 2andx = -5 - Example 2: One Solution
Equation:|x - 5| = 0
Input:a = 1,b = -5,c = 0
Output:x = 5 - Example 3: No Real Solutions (Negative C)
Equation:|3x + 1| = -2
Input:a = 3,b = 1,c = -2
Output: No real solutions. - Example 4: All Real Numbers (a=0 case)
Equation:|0x + 5| = 5(which is|5| = 5)
Input:a = 0,b = 5,c = 5
Output: All real numbers are solutions. - Example 5: No Real Solutions (a=0 case)
Equation:|0x + 5| = 3(which is|5| = 3)
Input:a = 0,b = 5,c = 3
Output: No real solutions.
By understanding these cases, you can effectively solve any absolute value equation of the form |ax + b| = c.