Linear Inequality Calculator

Linear Inequality Calculator

function calculateInequality() { var a = parseFloat(document.getElementById("coefficientA").value); var b = parseFloat(document.getElementById("constantB").value); var c = parseFloat(document.getElementById("constantC").value); var operator = document.getElementById("operator").value; var resultDiv = document.getElementById("result"); var solution = ""; if (isNaN(a) || isNaN(b) || isNaN(c)) { resultDiv.innerHTML = "Please enter valid numbers for all fields."; return; } // Original inequality: ax + b [operator] c // Step 1: Subtract b from both sides // ax [operator] c – b var rhs = c – b; if (a === 0) { // Special case: 0x [operator] rhs var isTrue = false; switch (operator) { case "<": isTrue = (0 ": isTrue = (0 > rhs); break; case "<=": isTrue = (0 =": isTrue = (0 >= rhs); break; } if (isTrue) { solution = "All real numbers"; } else { solution = "No solution"; } } else { // Step 2: Divide by a // x [new_operator] (c – b) / a var xValue = rhs / a; var newOperator = operator; if (a < 0) { // Flip the operator if dividing by a negative number switch (operator) { case ""; break; case ">": newOperator = "<"; break; case "="; break; case ">=": newOperator = "<="; break; } } solution = "x " + newOperator + " " + xValue.toFixed(4); // Display with 4 decimal places } resultDiv.innerHTML = "Solution: " + solution; }

Understanding Linear Inequalities

A linear inequality is a mathematical statement that compares two expressions using an inequality symbol. Unlike equations, which show that two expressions are equal, inequalities show that one expression is greater than, less than, greater than or equal to, or less than or equal to another expression.

The most common forms of linear inequalities with one variable (x) are:

• ax + b < c (less than)
• ax + b > c (greater than)
• ax + b ≤ c (less than or equal to)
• ax + b ≥ c (greater than or equal to)

Here, 'a', 'b', and 'c' are real numbers, and 'a' is typically not zero (though our calculator handles the a=0 case).

How to Solve Linear Inequalities

Solving a linear inequality means finding the range of values for the variable (usually 'x') that makes the inequality true. The process is very similar to solving linear equations, with one crucial difference:

1. Isolate the term with 'x': Just like with equations, your first step is to get the ax term by itself on one side of the inequality. This is done by adding or subtracting the constant 'b' from both sides.
   Example: If you have 2x + 3 < 11, subtract 3 from both sides: 2x < 11 - 3, which simplifies to 2x < 8.
2. Isolate 'x': Next, you need to get 'x' by itself. This is done by dividing (or multiplying) both sides by the coefficient 'a'.
   Example (continued): From 2x < 8, divide both sides by 2: x < 4.
3. The Critical Rule: Flipping the Inequality Sign: This is the most important difference from solving equations. If you multiply or divide both sides of an inequality by a negative number, you MUST reverse the direction of the inequality sign.
   Example: If you have -3x + 5 > 14:
   Subtract 5: -3x > 9
   Divide by -3 (and flip the sign): x < -3

Special Case: When the Coefficient 'a' is Zero

If the coefficient 'a' is zero, the inequality takes the form 0x + b [operator] c, which simplifies to b [operator] c. In this scenario, 'x' disappears, and you are left with a statement comparing two constants. There are two possible outcomes:

• All Real Numbers: If the resulting statement is true (e.g., 0 < 5 or 7 ≥ 7), then any real number for 'x' will satisfy the original inequality.
• No Solution: If the resulting statement is false (e.g., 0 > 5 or 3 < 2), then there is no real number for 'x' that can satisfy the original inequality.

Examples Using the Calculator

Let's walk through a few examples to see how the calculator works:

Example 1: Simple Inequality

• Inequality: 2x + 3 < 11
• Inputs: Coefficient 'a' = 2, Constant 'b' = 3, Operator = <, Constant 'c' = 11
• Calculation:
  1. 2x < 11 - 3
  2. 2x < 8
  3. x < 8 / 2
  4. x < 4
• Calculator Output: x < 4.0000

Example 2: Dividing by a Negative Number

• Inequality: -4x + 7 ≥ 19
• Inputs: Coefficient 'a' = -4, Constant 'b' = 7, Operator = ≥, Constant 'c' = 19
• Calculation:
  1. -4x ≥ 19 - 7
  2. -4x ≥ 12
  3. Divide by -4 and FLIP the sign: x ≤ 12 / -4
  4. x ≤ -3
• Calculator Output: x ≤ -3.0000

Example 3: Coefficient 'a' is Zero (All Real Numbers)

• Inequality: 0x + 5 < 10
• Inputs: Coefficient 'a' = 0, Constant 'b' = 5, Operator = <, Constant 'c' = 10
• Calculation:
  1. 0x < 10 - 5
  2. 0x < 5
  3. This simplifies to 0 < 5, which is TRUE.
• Calculator Output: All real numbers

Example 4: Coefficient 'a' is Zero (No Solution)

• Inequality: 0x + 8 > 15
• Inputs: Coefficient 'a' = 0, Constant 'b' = 8, Operator = >, Constant 'c' = 15
• Calculation:
  1. 0x > 15 - 8
  2. 0x > 7
  3. This simplifies to 0 > 7, which is FALSE.
• Calculator Output: No solution