Matematika Kelas 10 | Bahas Soal Pertidaksamaan linear satu variabel.

Introduction

This tutorial covers the topic of linear inequalities in one variable, specifically designed for 10th-grade mathematics. Understanding these concepts is crucial for solving real-world problems and preparing for advanced mathematical studies. By the end of this guide, you'll be equipped with the skills to tackle linear inequalities confidently.

Step 1: Understanding Linear Inequalities

• Definition: A linear inequality expresses a relationship where one side is not necessarily equal to the other. It can take forms like:
  • ( ax + b < c )
  • ( ax + b \leq c )
  • ( ax + b > c )
  • ( ax + b \geq c )
• Components:
  • Variable: The unknown quantity (e.g., ( x )).
  • Coefficient: The numerical factor multiplying the variable (e.g., ( a )).
  • Constant: The value without a variable (e.g., ( b ) or ( c )).

Practical Tip: Remember that solving an inequality is similar to solving an equation, but you must be careful when multiplying or dividing by a negative number, as it reverses the inequality sign.

Step 2: Solving Linear Inequalities

1. Isolate the variable:

   • Use addition or subtraction to move constants to the opposite side.
   • Use multiplication or division to isolate the variable.

   Example:

   • From ( 2x + 3 < 7 ):
     • Subtract 3: ( 2x < 4 )
     • Divide by 2: ( x < 2 )

2. Graphing the Solution:

   • Number Line: Represent the solution on a number line.
   • Use an open circle for < or >, and a closed circle for ≤ or ≥.
   • Shade the region that satisfies the inequality.

Common Pitfall: Always check your final answer by substituting the variable back into the original inequality to ensure it holds true.

Step 3: Working with Compound Inequalities

• Definition: A compound inequality combines two inequalities, often connected by "and" or "or".
• Examples:
  • And: ( a < x < b )
  • Or: ( x < a ) or ( x > b )

1. Solving Compound Inequalities:
   • Solve each part separately.
   • Graph the solution accordingly.

Real-World Application: Understanding compound inequalities can help in scenarios such as determining acceptable ranges for product prices or temperatures.

Step 4: Applications of Linear Inequalities

• Real Life Contexts: Linear inequalities can be used to model situations such as budgeting, where you want to keep expenses below a certain amount, or in physics to determine limits of speed or distance.

• Example: If you have a monthly budget of $500 and plan to spend $200 on groceries, you can represent your remaining budget as:

  • ( 200 + x < 500 )

Conclusion

In this tutorial, you learned how to understand and solve linear inequalities in one variable, including isolating the variable, graphing solutions, and dealing with compound inequalities. These skills are essential for both academic success in mathematics and real-world problem-solving.

As a next step, practice solving various linear inequalities and apply these concepts to real-life scenarios to reinforce your understanding.