Inequação do 1° grau com Fração

Introduction

This tutorial focuses on solving first-degree inequalities with fractions, a fundamental topic in mathematics often encountered in the seventh grade. Understanding how to manipulate and solve these inequalities is essential for mastering algebraic concepts and preparing for more advanced math.

Step 1: Understanding the Inequality

• An inequality compares two expressions, showing that one is greater than, less than, or equal to another.
• Common symbols include:
  • Greater than: >
  • Less than: <
  • Greater than or equal to: ≥
  • Less than or equal to: ≤

Step 2: Identifying the Fraction

• In a first-degree inequality involving fractions, identify the variable and the fraction involved.
• Example Inequality:
  • ( \frac{x}{2} + 3 < 5 )
• Here, ( x ) is the variable, and ( \frac{x}{2} ) is the fractional part.

Step 3: Isolating the Variable

• Move the constant term to the other side of the inequality to isolate the variable.
• Using the example from Step 2:
  1. Subtract 3 from both sides:
     • ( \frac{x}{2} < 5 - 3 )
     • Simplifying gives: ( \frac{x}{2} < 2 )

Step 4: Eliminating the Fraction

• To eliminate the fraction, multiply both sides of the inequality by the denominator (2 in this case).
• Ensure that the inequality sign stays the same since we are multiplying by a positive number:
  • ( x < 2 \times 2 )
  • This simplifies to ( x < 4 )

Step 5: Graphing the Solution

• Represent the solution on a number line:
  • Draw a number line and mark the point 4.
  • Use an open circle at 4 to indicate that 4 is not included in the solution (since it’s a strict inequality).
  • Shade the line to the left of 4 to show all values less than 4 are part of the solution.

Step 6: Checking the Solution

• It’s essential to check if your solution satisfies the original inequality.
• Choose a test value, such as 3:
  • Substitute into the original inequality:
  • ( \frac{3}{2} + 3 < 5 )
  • Simplifying gives ( 1.5 + 3 < 5 ) or ( 4.5 < 5 ), which is true.
• This confirms that the solution ( x < 4 ) is correct.

Conclusion

In this tutorial, we learned how to solve first-degree inequalities with fractions by isolating the variable, eliminating the fraction, and graphing the solution. Understanding these steps is crucial for tackling similar problems in algebra. Practice with various inequalities to strengthen your skills and ensure mastery of the topic.