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Solving Linear Inequalities: A Step-by-Step Guide

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Solving Linear Inequalities: A Step-by-Step Guide

Linear inequalities are mathematical statements that compare two expressions using inequality symbols such as <, >, ≤, or ≥. Solving linear inequalities involves finding the values of the variable that make the inequality true. This guide will provide a step-by-step approach to solving linear inequalities, ensuring you understand the process and can confidently tackle any problem.

Understanding the Basics

Before we dive into the steps, let's clarify some key concepts:

  • Inequality symbols:
    • < (less than)
    • > (greater than)
    • ≤ (less than or equal to)
    • ≥ (greater than or equal to)
  • Solution set: The set of all values of the variable that satisfy the inequality.
  • Graphing the solution: We represent the solution set on a number line.
  • Interval notation: A concise way to express the solution set using intervals.

Steps to Solve Linear Inequalities

Follow these steps to solve any linear inequality:

1. Simplify Both Sides

Begin by simplifying both sides of the inequality by combining like terms and removing any parentheses.

2. Isolate the Variable

Your goal is to get the variable by itself on one side of the inequality. To do this, perform the same operations on both sides, just like you would with an equation. Remember these key points:

  • Adding or subtracting: Adding or subtracting the same number from both sides does not change the inequality.
  • Multiplying or dividing by a positive number: Multiplying or dividing both sides by a positive number does not change the inequality.
  • Multiplying or dividing by a negative number: Multiplying or dividing both sides by a negative number reverses the inequality sign.

3. Express the Solution

Once you have isolated the variable, express the solution in one of the following ways:

  • Inequality notation: Write the inequality with the variable on one side and the solution on the other.
  • Graph: Represent the solution set on a number line. Use an open circle for < or > and a closed circle for ≤ or ≥.
  • Interval notation: Use parentheses for open intervals (not including endpoints) and square brackets for closed intervals (including endpoints).

Examples

Let's illustrate these steps with some examples:

Example 1:

Solve the inequality 2x + 5 < 11

  1. Simplify: The inequality is already simplified.
  2. Isolate x:
    • Subtract 5 from both sides: 2x < 6
    • Divide both sides by 2: x < 3
  3. Express the solution:
    • Inequality notation: x < 3
    • Graph: (Draw a number line with an open circle at 3 and shade to the left)
    • Interval notation: (-∞, 3)

Example 2:

Solve the inequality -3x + 6 ≥ 12

  1. Simplify: The inequality is already simplified.
  2. Isolate x:
    • Subtract 6 from both sides: -3x ≥ 6
    • Divide both sides by -3 (remember to reverse the inequality sign!): x ≤ -2
  3. Express the solution:
    • Inequality notation: x ≤ -2
    • Graph: (Draw a number line with a closed circle at -2 and shade to the left)
    • Interval notation: (-∞, -2]

Conclusion

Solving linear inequalities is a fundamental skill in algebra. By following the step-by-step guide outlined above, you can confidently solve any linear inequality and express the solution in different formats. Remember to pay attention to the inequality signs and the rules for multiplying or dividing by negative numbers. Practice these steps with various examples to solidify your understanding and become proficient in solving linear inequalities.

Inequalities