Solving Inequalities: Basic Rules and Techniques

Inequalities are a fundamental aspect of algebra, representing a relationship between two expressions that are not equal. While solving inequalities might seem similar to solving equations, there are specific rules to keep in mind to ensure the correct solution. This article will explore the basic rules of simplifying inequalities and demonstrate how to solve a sample inequality step by step.

Understanding Inequalities

Before diving into the solving process, it's important to understand what an inequality is. An inequality compares two expressions using symbols such as (less than), (less than or equal to), (greater than), and (greater than or equal to). For instance, consider the inequality:

frac12;w 3 11

Basic Rules for Simplifying Inequalities

Similar to solving equations, there are specific rules to simplify inequalities. These rules help to isolate the variable on one side of the inequality, leading to the solution. Here are the key rules:

Addition and Subtraction

When you add or subtract the same value from both sides of an inequality, the direction of the inequality remains the same. For example:

x - 2 5

Add 2 to both sides:

x - 2 2 5 2

x 7

Multiplication and Division by Positive Numbers

Multiplying or dividing both sides of an inequality by a positive number does not change the direction of the inequality. For example:

x / 2 4

Multiply both sides by 2:

x / 2 * 2 4 * 2

x 8

Multiplication and Division by Negative Numbers

When multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality is reversed. For example:

-2x 8

Divide both sides by -2 (and reverse the inequality sign):

-2x / -2 8 / -2

x -4

Solving a Sample Inequality

Let's break down the process of solving the inequality w/2 3 11.

Step 1: Eliminate the fraction

Multiply both sides by 2 to eliminate the fraction:

(w/2) * 2 3 * 2 11 * 2

w 6 22

Step 2: Isolate the variable

Subtract 6 from both sides to isolate the variable:

w 6 - 6 22 - 6

w 16

The solution to the inequality is w 16.

Conclusion

In conclusion, understanding the basic rules of simplifying inequalities is crucial for solving algebraic inequalities efficiently. By following the rules for addition, subtraction, multiplication, and division (with an emphasis on reversing the inequality sign when dealing with negative numbers), you can effectively isolate the variable and determine the solution. Applying these techniques to real-world problems can help in making more informed decisions and solving practical challenges.