How to Solve Inequalities Step-by-Step: A Comprehensive Guide

Understanding how to solve inequalities is a fundamental skill in mathematics. This guide will provide a detailed, step-by-step approach to solving various types of inequalities, including linear and quadratic ones. By mastering these operations, you can solve complex inequalities effectively.

Overview of Inequality Solving

Inequalities are mathematical statements that compare two expressions using symbols such as (less than), (greater than), le; (less than or equal to), and ge; (greater than or equal to). The process of solving inequalities is similar to solving equations, but with a few crucial differences in how certain operations affect the inequality sign.

Basic Principles for Solving Inequalities

When solving any inequality, the goal is to isolate the variable on one side of the inequality. Here are the key operations that affect the inequality sign:

Adding or Subtracting: Adding or subtracting the same number on both sides of the inequality does not change the direction of the sign. For example, if you subtract 3 from both sides of the inequality x 5, the sign remains the same. Multiplying or Dividing: Multiplying or dividing both sides of the inequality by a negative number will reverse the direction of the inequality sign. For example, if you multiply both sides of x 5 by -2, the inequality becomes -2x -10. Swapping Sides: Swapping the sides of an inequality will also change the direction of the sign. For example, if a b, then b a. Taking Reciprocals: Taking the reciprocal of both sides of the inequality will reverse the inequality sign. For example, if a b, then 1/a 1/b. Applying Decreasing Functions: Applying a decreasing function to both sides of the inequality will reverse the inequality sign. For example, if a b, then log(a) log(b), assuming the logarithmic function is decreasing.

Examples of Solving Linear Inequalities

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

Example 1

Solve the inequality x 4.

The inequality x 4 is already in its simplest form. There's no need for further solving since x is already isolated on one side.

Example 2

Solve the inequality x - 1 5.

To solve this, add 1 to both sides of the inequality:

x - 1 1 5 1 (Adding 1 to both sides)

x 6

Therefore, the solution is x 6.

Example 3

Solve the inequality x 2 ge; 7.

To solve this, subtract 2 from both sides of the inequality:

x 2 - 2 ge; 7 - 2 (Subtracting 2 from both sides)

x ge; 5

Therefore, the solution is x ge; 5.

Examples of Solving Quadratic Inequalities

Quadratic inequalities involve a second-degree polynomial, and their solutions are often intervals rather than single values. Let's consider an example:

Example 4

Solve the inequality x^2 - 4x 3 0.

First, solve the corresponding equation:

x^2 - 4x 3 0

This factors into:

(x - 1)(x - 3) 0

Setting each factor to zero gives:

x 1 or x 3

These roots divide the number line into three intervals: (-∞, 1), (1, 3), and (3, ∞). To determine where the inequality x^2 - 4x 3 0 holds, test a point in each interval:

For x 0 in (-∞, 1), (0 - 1)(0 - 3) (-1)(-3) 3 0, so x^2 - 4x 3 0. For x 2 in (1, 3), (2 - 1)(2 - 3) (1)(-1) -1 0, so x^2 - 4x 3 0. For x 4 in (3, ∞), (4 - 1)(4 - 3) (3)(1) 3 0, so x^2 - 4x 3 0.

Therefore, the solution to the inequality x^2 - 4x 3 0 is (-∞, 1) ∪ (3, ∞).

Conclusion

Mastering the art of solving inequalities is crucial for tackling more complex mathematical problems. By carefully considering the operations and their impact on the inequality sign, you can solve a wide range of inequalities effectively. Whether dealing with linear or quadratic inequalities, the key is to isolate the variable and apply the appropriate operations to maintain the inequality's direction.