#3 Sistem Bilangan | KALKULUS

Introduction

In this tutorial, we will explore the concepts of quadratic inequalities, fractional inequalities, number lines, and positive-negative graphs as discussed in the video from Study With Student. These concepts are essential in calculus and will help you understand how to solve inequalities and visualize their solutions effectively.

Step 1: Understanding Quadratic Inequalities

• A quadratic inequality is an inequality that involves a quadratic expression.
• To solve a quadratic inequality, follow these steps:
  1. Set the inequality to zero: For example, if you have ( ax^2 + bx + c < 0 ), rearrange it to ( ax^2 + bx + c = 0 ).
  2. Find the roots: Use the quadratic formula ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ) to calculate the roots.
  3. Test intervals: Divide the number line into intervals based on the roots and test a point from each interval to determine where the inequality holds true.

Step 2: Solving Fractional Inequalities

• Fractional inequalities involve a rational expression. To solve these:
  1. Identify the critical points: Set the numerator and denominator to zero to find critical points.
  2. Create a sign chart: Determine the sign of the expression in each interval created by these points.
  3. Write the solution set: Based on your sign chart, express the solution in interval notation.

Step 3: Using the Number Line

• A number line is a visual tool that helps to understand where the solutions to inequalities lie.
• To effectively use a number line:
  1. Plot critical points: Mark the roots and critical points on the number line.
  2. Shade intervals: Based on your solutions from previous steps, shade the regions that satisfy the inequality.
  3. Indicate open and closed intervals: Use open circles for non-inclusive boundaries and closed circles for inclusive boundaries.

Step 4: Graphing Positive and Negative Values

• Understanding how to graph positive and negative values is essential for visualizing inequalities.
• Follow these steps to graph:
  1. Choose a test point: Select a simple value (like 0) to see if it satisfies the inequality.
  2. Plot points: For each interval, plot points corresponding to the quadratic or fractional function.
  3. Draw the graph: Connect the points smoothly and use dashed lines for inequalities that do not include the boundary.

Conclusion

In this tutorial, we covered the essentials of quadratic and fractional inequalities, how to analyze and plot their solutions on a number line, and how to graph the corresponding functions. Understanding these concepts will enhance your problem-solving skills in calculus. As a next step, practice solving various inequalities and plot their solutions to reinforce your learning. For further exploration, consider downloading the provided slide materials for additional insights and examples.