Sistem Pertidaksamaan Linear

Introduction

This tutorial provides a comprehensive guide to understanding linear inequalities, a fundamental topic in mathematics, particularly for 11th-grade students. By the end of this tutorial, you will learn how to identify, graph, and solve linear inequalities, enhancing your mathematical skills for both academic and practical applications.

Step 1: Understanding Linear Inequalities

Linear inequalities are similar to linear equations but involve an inequality sign (>, <, ≥, ≤) instead of an equal sign. They represent a range of values rather than a single solution.

Key Concepts

• Inequality Symbols:
  • > means greater than
  • < means less than
  • ≥ means greater than or equal to
  • ≤ means less than or equal to
• Solution Set: The set of all possible values that satisfy the inequality.

Step 2: Solving Linear Inequalities

To solve a linear inequality, follow these steps:

1. Isolate the Variable:

   • Just like solving an equation, you want to get the variable (usually x) on one side.
   • Example: For the inequality (2x + 3 < 7), subtract 3 from both sides to get (2x < 4).

2. Divide or Multiply:

   • If you need to divide or multiply by a negative number, remember to reverse the inequality sign.
   • Example: From (2x < 4), divide both sides by 2 to get (x < 2).

3. Write the Solution:

   • Solutions can be expressed in interval notation or on a number line.
   • Example: The solution (x < 2) can be written in interval notation as ((-∞, 2)).

Step 3: Graphing Linear Inequalities

Graphing helps visualize the solutions of linear inequalities.

1. Draw the Coordinate Plane:

   • Label the x-axis and y-axis.

2. Graph the Boundary Line:

   • For (y < mx + b), draw a dashed line if the inequality is strict (>, <) and a solid line for non-strict (≥, ≤).
   • Example: For the inequality (y ≤ 2x + 3), draw a solid line for (y = 2x + 3).

3. Shade the Appropriate Area:

   • Shade above the line for (y > mx + b) and below for (y < mx + b).
   • This shaded area represents all the solutions to the inequality.

Step 4: Solving Systems of Linear Inequalities

When dealing with multiple inequalities, follow these steps:

1. Graph Each Inequality:

   • Use the steps from Step 3 for each inequality in the system.

2. Identify the Overlapping Region:

   • The solution set is where the shaded areas overlap.

3. Write the Final Solution:

   • Present the solution in terms of the overlapping region on the graph.

Conclusion

In this tutorial, you learned the basics of linear inequalities, how to solve them, graph them, and tackle systems of linear inequalities. Mastering these concepts is crucial for your mathematical development. As a next step, practice solving various inequalities and graphing them to reinforce your understanding. Consider exploring related topics such as linear programming for real-world applications.