Inequality Calculator Step by Step

Linear Inequality Solver

Enter the components of your linear inequality in the form ax + b [sign] c:

Original Inequality:

Step 1:

Step 2:

Step 2:

Step 2:

Final Solution:

Understanding and Solving Linear Inequalities Step-by-Step

Linear inequalities are mathematical statements that compare two expressions using an inequality symbol rather than an equality sign. Unlike equations, which typically have a single solution, inequalities often have a range of solutions. Understanding how to solve them is fundamental in algebra and various real-world applications, from budgeting to engineering.

What is a Linear Inequality?

A linear inequality involves a linear expression (an expression where the highest power of the variable is 1) on one or both sides of an inequality symbol. The common inequality symbols are:

• < (less than)
• > (greater than)
• ≤ (less than or equal to)
• ≥ (greater than or equal to)

A typical linear inequality takes the form ax + b < c (or with any other inequality sign), where 'a', 'b', and 'c' are constants, and 'x' is the variable.

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.
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.
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 students make.

Step-by-Step Process to Solve ax + b [sign] c

Let's break down the process using the standard form ax + b [sign] c:

1. Isolate the term with 'x': Your first goal is to get the ax term by itself on one side of the inequality. To do this, subtract the constant 'b' from both sides of the inequality.
   ax + b - b [sign] c - b
ax [sign] c - b
2. Isolate 'x': Next, you need to get 'x' by itself. Divide both sides of the inequality by the coefficient 'a'.
   • If 'a' is a positive number, the inequality sign remains the same.
     x [sign] (c - b) / a
   • If 'a' is a negative number, you must flip the direction of the inequality sign.
     x [flipped sign] (c - b) / a
   • If 'a' is zero (0x + b [sign] c), the 'x' term disappears. You are left with b [sign] c.
     • If this resulting statement is true (e.g., 5 < 10), then all real numbers are solutions.
     • If this resulting statement is false (e.g., 10 < 5), then there is no solution.

Using the Inequality Solver

Our Linear Inequality Solver simplifies this process for you. Here's how to use it:

1. Coefficient of x (a): Enter the number that multiplies 'x'. For example, in 2x + 5 < 15, 'a' is 2.
2. Constant on left side (b): Enter the constant term added or subtracted on the left side. In 2x + 5 < 15, 'b' is 5.
3. Inequality Sign: Select the appropriate inequality symbol from the dropdown menu (<, >, ≤, ≥).
4. Constant on right side (c): Enter the constant term on the right side of the inequality. In 2x + 5 < 15, 'c' is 15.
5. Click "Solve Inequality" to see the final solution and a detailed step-by-step breakdown of how it was reached.

Examples with Realistic Numbers:

Example 1: Simple Positive Coefficient

Solve: 3x + 7 > 22

• a = 3, b = 7, c = 22, sign = >
• Step 1: Subtract 7 from both sides:
  3x > 22 - 7
3x > 15
• Step 2: Divide by 3 (positive, so sign stays):
  x > 15 / 3
x > 5

Example 2: Negative Coefficient (Sign Flip)

Solve: -2x + 10 ≤ 4

• a = -2, b = 10, c = 4, sign = ≤
• Step 1: Subtract 10 from both sides:
  -2x ≤ 4 - 10
-2x ≤ -6
• Step 2: Divide by -2 (negative, so sign flips from ≤ to ≥):
  x ≥ -6 / -2
x ≥ 3

Example 3: Coefficient of x is Zero

Solve: 0x + 8 < 5

• a = 0, b = 8, c = 5, sign = <
• Step 1: Subtract 8 from both sides:
  0x < 5 - 8
0x < -3
• Step 2: The 'x' term vanishes, leaving:
  8 < 5
• This statement is FALSE. Therefore, there is no solution.

By following these steps and using the calculator, you can confidently solve a wide range of linear inequalities and understand the reasoning behind each step.