How to Solve a Linear Inequality?

Solving a linear inequality follows a similar process to solving a linear equation.

Steps to Solve Linear Inequalities

The core idea is to isolate the variable on one side of the inequality. Here's a breakdown of the steps involved, drawing from the provided reference:

1. Simplify Both Sides: Like solving equations, start by simplifying both sides of the inequality. This involves:

   • Combining like terms on each side.
   • Using the distributive property to remove parentheses.

2. Isolate the Variable Terms: The goal is to get all terms containing the variable on one side of the inequality and all the constant terms (numbers) on the other side. This is achieved using inverse operations:

   • Add or subtract the same term from both sides of the inequality to move variable terms and constants to their respective sides.
   • Remember: Performing the same operation on both sides maintains the inequality.

3. Multiply or Divide by the Coefficient: After isolating the variable term, you will typically have a coefficient multiplying the variable.

   • Divide both sides of the inequality by the coefficient to isolate the variable entirely.
   • Critical Rule: If you multiply or divide both sides by a negative number, you must reverse the direction of the inequality sign.

4. Express the Solution: The solution will express all possible values that satisfy the inequality. This is typically written in one of three ways:

   • Using inequality notation, e.g., x < 5 or y ≥ -2.
   • Graphically on a number line.
   • In interval notation.

Example

Let’s go through an example:

2x + 3 < 7

1. Simplify: Both sides are already simplified.
2. Isolate Variable Term: Subtract 3 from both sides:
   2x < 4
3. Divide by Coefficient: Divide both sides by 2:
   x < 2

Key Differences from Solving Equations

While similar, there's one crucial difference between solving inequalities and solving equations:

• Reversing the Inequality Sign: Whenever 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 had -x > 5 and multiplied both sides by -1, the inequality becomes x < -5

Summary of Steps

Step	Description
1. Simplify	Combine like terms and eliminate parentheses on each side.
2. Isolate Variable Terms	Move variable terms to one side and constant terms to the other by adding or subtracting from both sides.
3. Multiply/Divide	Divide both sides by the variable's coefficient. Remember to reverse the inequality sign if you divide by a negative number.
4. Express Solution	Write the solution set using inequality notation, on a number line, or using interval notation.

By following these steps, you can confidently solve a wide range of linear inequalities.