Inequality Calculator Graph

Linear Inequality Solver

This calculator helps you solve linear inequalities of the form ax + b [operator] c and provides a description of how to represent the solution graphically on a number line.

">> (Greater Than) <option value="< (Less Than) =" selected>≥ (Greater Than or Equal To) <option value="≤ (Less Than or Equal To)

Solution:

${originalInequality}

${solutionText}

Graphical Interpretation (Number Line):

Understanding and Graphing Linear Inequalities

Linear inequalities are fundamental concepts in algebra that extend the idea of linear equations. While a linear equation like ax + b = c has a single solution (or no solution, or infinitely many solutions in special cases), a linear inequality typically has a range of solutions, which can be represented as an interval on a number line.

What is a Linear Inequality?

A linear inequality is a mathematical statement that compares two expressions using an inequality symbol. The general form is ax + b [operator] c, where:

• a and b are coefficients (real numbers), with a not equal to zero for a standard linear inequality.
• x is the variable.
• c is a constant (real number).
• [operator] can be one of four symbols:
  • > (greater than)
  • < (less than)
  • ≥ (greater than or equal to)
  • ≤ (less than or equal to)

Unlike equations, which seek a specific value that makes the statement true, inequalities seek a set of values (an interval) that satisfy the condition.

How to Solve Linear Inequalities

Solving linear inequalities is very similar to solving linear equations, with one crucial difference:

1. Isolate the variable term: Use addition or subtraction to move all constant terms to one side of the inequality and terms with the variable to the other. For example, in ax + b > c, subtract b from both sides to get ax > c - b.
2. Isolate the variable: Divide or multiply both sides by the coefficient of the variable (a in our example).
3. The Critical Rule: If you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. For example, if you have -2x > 6 and divide by -2, it becomes x < -3.

Special Cases:

• If a = 0: The inequality simplifies to b [operator] c.
  • If this resulting statement is true (e.g., 5 > 2), then the solution is "All real numbers" (any value of x will satisfy it).
  • If this resulting statement is false (e.g., 5 < 2), then there is "No solution" (no value of x will satisfy it).

Graphing Linear Inequalities on a Number Line

The solution to a one-variable linear inequality is an interval, which can be visually represented on a number line. Here's how:

1. Find the Critical Point: This is the value of x that makes the inequality an equality (e.g., if x > 5, the critical point is 5).
2. Determine the Type of Circle:
   • Use an open circle (○) at the critical point if the inequality is strict (> or <). This indicates that the critical point itself is NOT part of the solution.
   • Use a closed circle (●) at the critical point if the inequality is non-strict (≥ or ≤). This indicates that the critical point IS part of the solution.
3. Determine the Direction of Shading:
   • If the inequality is > or ≥, shade the number line to the right of the critical point.
   • If the inequality is < or ≤, shade the number line to the left of the critical point.

How to Use the Linear Inequality Solver

Our calculator simplifies the process of solving and understanding linear inequalities:

1. Coefficient 'a': Enter the number multiplying x.
2. Coefficient 'b': Enter the constant term on the left side of the inequality.
3. Inequality Operator: Select the appropriate comparison symbol (>, <, ≥, or ≤).
4. Constant 'c': Enter the constant term on the right side of the inequality.
5. Click "Solve Inequality" to see the solution set for x and a description of its graphical representation on a number line.

Examples:

Example 1: Simple Greater Than

Inequality: 2x + 3 > 11

• Input: a=2, b=3, operator=>, c=11
• Calculation:
  1. 2x > 11 - 3
  2. 2x > 8
  3. x > 4 (divide by positive 2, so operator doesn't change)
• Solution: x > 4
• Graph: An open circle at 4, shaded to the right.

Example 2: Less Than or Equal To with Negative Coefficient

Inequality: -3x + 5 ≤ 14

• Input: a=-3, b=5, operator=≤, c=14
• Calculation:
  1. -3x ≤ 14 - 5
  2. -3x ≤ 9
  3. x ≥ -3 (divide by negative 3, so operator reverses from ≤ to ≥)
• Solution: x ≥ -3
• Graph: A closed circle at -3, shaded to the right.

Example 3: No Solution Case

Inequality: 0x + 7 < 2

• Input: a=0, b=7, operator=<, c=2
• Calculation:
  1. Simplifies to 7 < 2
  2. This statement is false.
• Solution: No solution
• Graph: No points on the number line satisfy the inequality.

Example 4: All Real Numbers Case

Inequality: 0x + 1 ≥ 1

• Input: a=0, b=1, operator=≥, c=1
• Calculation:
  1. Simplifies to 1 ≥ 1
  2. This statement is true.
• Solution: All real numbers
• Graph: The entire number line is shaded.