Compound Inequalities Calculator

Compound Inequalities Calculator

Use this calculator to find the solution set for common types of compound inequalities involving two critical values.

Understanding Compound Inequalities

A compound inequality combines two simple inequalities with either the word "and" or the word "or." These inequalities help describe a range of values that a variable can take.

"AND" Compound Inequalities (Intersection)

When two inequalities are joined by "and," the solution set includes all values that satisfy both inequalities simultaneously. This often results in a single interval where the variable is "between" two numbers.

• Example: x > 5 AND x < 10
• This means x must be greater than 5 AND less than 10. The solution is all numbers between 5 and 10, not including 5 or 10.
• Inequality Notation: 5 < x < 10
• Interval Notation: (5, 10)

If there is no overlap between the two inequalities, an "and" compound inequality will have "No Solution." For example, x < 5 AND x > 10 has no solution because a number cannot be both less than 5 and greater than 10 at the same time.

"OR" Compound Inequalities (Union)

When two inequalities are joined by "or," the solution set includes all values that satisfy at least one of the inequalities. This typically results in two separate intervals, often described as the variable being "outside" two numbers.

• Example: x < 5 OR x > 10
• This means x can be any number less than 5, OR any number greater than 10.
• Inequality Notation: x < 5 OR x > 10
• Interval Notation: (-∞, 5) ∪ (10, ∞)

In some cases, an "or" compound inequality might cover all real numbers. For example, x < 10 OR x > 5 would mean all real numbers, as any number is either less than 10 or greater than 5 (or both).

How to Use the Calculator

1. Enter First Critical Value (A): Input the first number that defines a boundary for your inequality.
2. Enter Second Critical Value (B): Input the second number that defines a boundary.
3. Select Compound Inequality Type: Choose the type that matches your problem. The calculator will automatically handle the order of A and B for the solution.
4. Click "Calculate Solution": The calculator will display the original inequality, its solution in inequality notation, and its solution in interval notation.

Examples of Compound Inequalities

Let's look at some practical examples:

• Example 1: Temperature Range
  A comfortable room temperature is between 68 and 72 degrees Fahrenheit, inclusive.
  68 ≤ T ≤ 72
  Using the calculator: A=68, B=72, Type: A ≤ x ≤ B.
  Solution: [68, 72]
• Example 2: Speed Limits
  On a certain road, the speed limit is 60 mph, and you must drive at least 30 mph.
  30 ≤ S ≤ 60
  Using the calculator: A=30, B=60, Type: A ≤ x ≤ B.
  Solution: [30, 60]
• Example 3: Out-of-Range Values
  A machine operates correctly if its pressure is less than 10 PSI or greater than 50 PSI.
  P < 10 OR P > 50
  Using the calculator: A=10, B=50, Type: x < A OR x > B.
  Solution: (-∞, 10) ∪ (50, ∞)
• Example 4: No Solution Case
  If you try to find a number that is greater than 10 AND less than 5.
  x > 10 AND x < 5
  Using the calculator: A=10, B=5, Type: A < x < B (or A ≤ x ≤ B).
  Solution: No Solution
• Example 5: All Real Numbers Case
  If you try to find a number that is less than 10 OR greater than 5.
  x < 10 OR x > 5
  Using the calculator: A=10, B=5, Type: x < A OR x > B (the calculator will reorder to x < B OR x > A).
  Solution: (-∞, ∞)

= numB) { solutionInequalityText = "No Solution"; solutionIntervalText = "∅"; } else { solutionInequalityText = numA + " < x < " + numB; solutionIntervalText = "(" + numA + ", " + numB + ")"; } } else if (inequalityType === "A_le_x_le_B") { // A <= x numB) { solutionInequalityText = "No Solution"; solutionIntervalText = "∅"; } else if (numA === numB) { solutionInequalityText = "x = " + numA; solutionIntervalText = "[" + numA + "]"; } else { solutionInequalityText = numA + " ≤ x ≤ " + numB; solutionIntervalText = "[" + numA + ", " + numB + "]"; } } else if (inequalityType === "x_lt_A_or_x_gt_B") { // x B if (numA === numB) { solutionInequalityText = "x ≠ " + numA; solutionIntervalText = "(-∞, " + numA + ") ∪ (" + numA + ", ∞)"; } else if (numA > numB) { // e.g., x 5 -> x 10 solutionInequalityText = "x " + numA; solutionIntervalText = "(-∞, " + numB + ") ∪ (" + numA + ", ∞)"; } else { // numA < numB solutionInequalityText = "x " + numB; solutionIntervalText = "(-∞, " + numA + ") ∪ (" + numB + ", ∞)"; } } else if (inequalityType === "x_le_A_or_x_ge_B") { // x = B if (numA === numB) { solutionInequalityText = "All Real Numbers"; solutionIntervalText = "(-∞, ∞)"; } else if (numA > numB) { // e.g., x = 5 -> All Real Numbers solutionInequalityText = "All Real Numbers"; solutionIntervalText = "(-∞, ∞)"; } else { // numA < numB solutionInequalityText = "x ≤ " + numA + " OR x ≥ " + numB; solutionIntervalText = "(-∞, " + numA + "] ∪ [" + numB + ", ∞)"; } } resultDiv.innerHTML = "Original Inequality: " + originalInequalityText + "" + "Solution (Inequality Notation): " + solutionInequalityText + "" + "Solution (Interval Notation): " + solutionIntervalText + ""; }