Solve for Inequality Calculator

Linear Inequality Solver

Enter the coefficients and constants for a linear inequality of the form ax + b [operator] c.

Understanding and Solving Linear Inequalities

Inequalities are mathematical statements that compare 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. They are fundamental in various fields, from economics to engineering, for defining ranges and constraints.

What is a Linear Inequality?

A linear inequality is an inequality that involves a linear expression. This means the highest power of the variable (usually 'x') is 1. They typically take forms such as:

• ax + b < c
• ax + b > c
• ax + b ≤ c
• ax + b ≥ c

Where 'a', 'b', and 'c' are real numbers, and 'a' is not necessarily zero.

Key Rules for Solving Linear Inequalities

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

1. Adding or Subtracting: You can add or subtract the same number from both sides of an inequality without changing the direction of the inequality sign.
   Example: If x - 3 < 5, then x - 3 + 3 < 5 + 3, which means x < 8.
2. Multiplying or Dividing by a Positive Number: You can multiply or divide both sides of an inequality by the same positive number without changing the direction of the inequality sign.
   Example: If 2x < 10, then 2x / 2 < 10 / 2, which means x < 5.
3. Multiplying or Dividing by a Negative Number: If you multiply or divide both sides of an inequality by a negative number, you MUST REVERSE THE DIRECTION OF THE INEQUALITY SIGN. This is the most common mistake when solving inequalities.
   Example: If -3x < 12, then -3x / -3 > 12 / -3, which means x > -4. (Notice the sign flipped from '<' to '>')

Steps to Solve a Linear Inequality (e.g., ax + b < c)

1. Isolate the term with 'x': Subtract 'b' from both sides of the inequality.
   ax < c - b
2. Isolate 'x': Divide both sides by 'a'.
   • If 'a' is positive, the inequality sign remains the same: x < (c - b) / a
   • If 'a' is negative, reverse the inequality sign: x > (c - b) / a
3. Special Case (a = 0): If 'a' is zero, the 'x' term disappears, leaving b < c.
   • If b < c is true (e.g., 5 < 10), then the inequality is true for all real numbers.
   • If b < c is false (e.g., 10 < 5), then there is no solution.

Examples:

Let's walk through a few examples to illustrate the process:

Example 1: Simple Inequality

Solve: 3x + 7 < 19

1. Subtract 7 from both sides: 3x < 19 - 7 3x < 12
2. Divide by 3 (positive number, sign stays the same): x < 12 / 3 x < 4
3. Solution: x < 4

Example 2: Inequality with a Negative Coefficient

Solve: -2x + 1 ≥ 9

1. Subtract 1 from both sides: -2x ≥ 9 - 1 -2x ≥ 8
2. Divide by -2 (negative number, FLIP THE SIGN): x ≤ 8 / -2 x ≤ -4
3. Solution: x ≤ -4

Example 3: Special Case (a = 0)

Solve: 0x + 5 < 10

1. The 'x' term vanishes: 5 < 10
2. Is 5 < 10 true? Yes.
3. Solution: All real numbers.

Solve: 0x + 12 > 20

1. The 'x' term vanishes: 12 > 20
2. Is 12 > 20 true? No.
3. Solution: 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:

• Coefficient 'a': The number multiplying 'x'.
• Constant 'b': The number added to or subtracted from 'ax'.
• Operator: Choose the correct comparison symbol (<, >, ≤, ≥).
• Constant 'c': The number on the right side of the inequality.

Click "Solve Inequality," and the calculator will instantly provide the solution for 'x', handling all the rules, including sign flipping for negative coefficients and special cases where 'a' is zero.