Solving Inequalities Calculator with Steps

Linear Inequality Solver

Enter the number 'a' in the form ax + b [operator] c.
Enter the number 'b' in the form ax + b [operator] c.
<option value="< ">> <option value="≤ =" selected>≥ Choose the inequality symbol.
Enter the number 'c' in the form ax + b [operator] c.

Solution Steps:

Enter values and click "Solve Inequality" to see the steps.

Final Solution:

The solution will appear here.

function calculateInequality() { var coeffA = parseFloat(document.getElementById("coeffA").value); var constB = parseFloat(document.getElementById("constB").value); var operator = document.getElementById("operator").value; var constC = parseFloat(document.getElementById("constC").value); var solutionStepsDiv = document.getElementById("solutionSteps"); var finalSolutionDiv = document.getElementById("finalSolution"); var steps = []; var finalSolution = ""; // Input validation if (isNaN(coeffA) || isNaN(constB) || isNaN(constC)) { solutionStepsDiv.innerHTML = '

Solution Steps:

Please enter valid numbers for all fields.'; finalSolutionDiv.innerHTML = '

Final Solution:

Error: Invalid input.'; return; } // Helper function to format numbers function formatNum(num) { return num % 1 === 0 ? num.toString() : num.toFixed(4); } // Initial inequality steps.push("1. Original inequality: " + formatNum(coeffA) + "x + " + formatNum(constB) + " " + operator + " " + formatNum(constC)); // Handle case where coefficient 'a' is zero if (coeffA === 0) { var result = false; switch (operator) { case "<": result = (constB ": result = (constB > constC); break; case "<=": result = (constB =": result = (constB >= constC); break; } steps.push("2. Simplify: " + formatNum(constB) + " " + operator + " " + formatNum(constC)); if (result) { finalSolution = "All real numbers (True statement)"; steps.push("3. Since " + formatNum(constB) + " " + operator + " " + formatNum(constC) + " is true, the solution is all real numbers."); } else { finalSolution = "No solution (False statement)"; steps.push("3. Since " + formatNum(constB) + " " + operator + " " + formatNum(constC) + " is false, there is no solution."); } } else { // Step 2: Subtract 'b' from both sides var rightSideAfterB = constC – constB; steps.push("2. Subtract " + formatNum(constB) + " from both sides: " + formatNum(coeffA) + "x " + operator + " " + formatNum(constC) + " – " + formatNum(constB)); steps.push("3. Simplify: " + formatNum(coeffA) + "x " + operator + " " + formatNum(rightSideAfterB)); // Step 3: Divide by 'a' var finalOperator = operator; if (coeffA < 0) { // Flip the operator if dividing by a negative number switch (operator) { case ""; break; case ">": finalOperator = "<"; break; case "="; break; case ">=": finalOperator = "<="; break; } steps.push("4. Divide both sides by " + formatNum(coeffA) + " and reverse the inequality sign:"); } else { steps.push("4. Divide both sides by " + formatNum(coeffA) + ":"); } var solutionValue = rightSideAfterB / coeffA; steps.push("   x " + finalOperator + " " + formatNum(rightSideAfterB) + " / " + formatNum(coeffA)); steps.push("5. Simplify: x " + finalOperator + " " + formatNum(solutionValue)); finalSolution = "x " + finalOperator + " " + formatNum(solutionValue); } solutionStepsDiv.innerHTML = '

Solution Steps:

' + steps.map(function(step) { return " + step + "; }).join("); finalSolutionDiv.innerHTML = '

Final Solution:

' + finalSolution + "; }

Understanding and Solving 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 solution to a linear inequality is typically a range of values, rather than a single value.

The Basics of Linear Inequalities

A common form of a linear inequality is ax + b < c, where:

  • x is the variable.
  • a, b, and c are constant numbers.
  • The inequality symbol can be < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to).

Solving linear inequalities involves isolating the variable, similar to solving linear equations, but with one crucial difference: when you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign.

Steps to Solve a Linear Inequality

  1. Simplify Both Sides: If necessary, distribute, combine like terms, and simplify both sides of the inequality.
  2. Isolate the Variable Term: Use addition or subtraction to move all terms containing the variable to one side of the inequality and all constant terms to the other side.
  3. Isolate the Variable: Use multiplication or division to get the variable by itself.
  4. Remember the Rule for Negative Numbers: If you multiply or divide both sides of the inequality by a negative number, you MUST reverse the inequality sign. For example, < becomes >, and ≥ becomes ≤.
  5. Express the Solution: The solution will be in the form x < k, x > k, x ≤ k, or x ≥ k, where k is a constant.

Examples of Solving Linear Inequalities

Example 1: Positive Coefficient

Let's solve the inequality: 3x + 7 > 19

  1. Original inequality: 3x + 7 > 19
  2. Subtract 7 from both sides: 3x > 19 - 7
  3. Simplify: 3x > 12
  4. Divide both sides by 3 (positive number, so no sign flip): x > 12 / 3
  5. Final Solution: x > 4

Example 2: Negative Coefficient (Sign Flip)

Let's solve the inequality: -2x + 5 ≤ 15

  1. Original inequality: -2x + 5 ≤ 15
  2. Subtract 5 from both sides: -2x ≤ 15 - 5
  3. Simplify: -2x ≤ 10
  4. Divide both sides by -2 (negative number, so flip the sign): x ≥ 10 / -2
  5. Final Solution: x ≥ -5

Example 3: When the Variable Coefficient is Zero

Consider the inequality: 0x + 5 < 10

  1. Original inequality: 0x + 5 < 10
  2. Simplify: 5 < 10
  3. Since 5 < 10 is a true statement, the solution is All real numbers.

Consider the inequality: 0x + 10 < 5

  1. Original inequality: 0x + 10 < 5
  2. Simplify: 10 < 5
  3. Since 10 < 5 is a false statement, there is No solution.

How to Use the Linear Inequality Solver

Our calculator simplifies the process of solving linear inequalities of the form ax + b [operator] c. Simply input the values for a (coefficient of x), b (constant term on the left), and c (constant term on the right). Select the appropriate inequality operator (<, >, ≤, or ≥) from the dropdown menu. Click "Solve Inequality" to instantly see the step-by-step solution and the final answer.

This tool is perfect for students learning algebra, or anyone needing a quick verification of their inequality solutions.

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