Solving Linear Inequalities: A Step-by-Step Guide

In the realm of mathematics, linear inequalities play a crucial role in representing relationships where one quantity is either greater than, less than, greater than or equal to, or less than or equal to another quantity. Solving linear inequalities involves finding the values of the variable that satisfy the given inequality. This guide will provide a step-by-step approach to solving linear inequalities, ensuring you master this essential concept.

Understanding Linear Inequalities

Linear inequalities are mathematical expressions that compare two quantities using inequality symbols such as:

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

For example, 2x + 3 > 7 is a linear inequality. The variable x is unknown, and we need to find its values that satisfy the inequality.

Steps to Solve Linear Inequalities

Solving linear inequalities follows a similar process to solving linear equations, with a few key differences. Here's a step-by-step guide:

1. Simplify both sides: Combine like terms and simplify both sides of the inequality as much as possible.
2. Isolate the variable: Use inverse operations (addition, subtraction, multiplication, division) to isolate the variable on one side of the inequality. Remember that if you multiply or divide both sides by a negative number, you must reverse the direction of the inequality symbol.
3. Express the solution: Write the solution in interval notation or on a number line. Interval notation is a concise way to represent a range of values. For example, the interval (2, 5) represents all numbers greater than 2 and less than 5. When graphing on a number line, use an open circle for strict inequalities (>, <) and a closed circle for inequalities with an equal sign (≥, ≤).

Examples

Let's illustrate the process with some examples:

Example 1:

Solve the inequality 3x - 5 ≤ 10.

1. Add 5 to both sides: 3x ≤ 15
2. Divide both sides by 3: x ≤ 5

The solution is x ≤ 5, which can be represented in interval notation as (-∞, 5] and on a number line with a closed circle at 5 and an arrow pointing to the left.

Example 2:

Solve the inequality -2x + 4 > 8.

1. Subtract 4 from both sides: -2x > 4
2. Divide both sides by -2 (remember to reverse the inequality sign): x < -2

The solution is x < -2, which can be represented in interval notation as (-∞, -2) and on a number line with an open circle at -2 and an arrow pointing to the left.

Conclusion

Solving linear inequalities is a fundamental skill in algebra and has applications in various fields, including economics, physics, and engineering. By following the steps outlined in this guide, you can confidently solve linear inequalities and understand their implications.