Guide to Solving Equations Involving Fractions and Variables

Equations often appear in various forms, especially those involving fractions and variables. Understanding how to solve such equations is crucial for mathematicians, engineers, and anyone dealing with basic algebra. In this guide, we will walk through a step-by-step process to solve an equation with fractions and variables, making it clear and easy to follow.

Solving the Equation: 0.5x - 3 0.75x 2

Let's start with the equation:

0.5x - 3 0.75x 2

The goal is to isolate the variable x. Let's break down the process into clear, manageable steps.

Step 1: Eliminate the Fractions

To make the calculations simpler, we need to eliminate the fractions. The least common multiple (LCM) of the denominators (0.5 and 0.75) is 4. Multiply the entire equation by 4:

4(0.5x - 3) 4(0.75x 2)

This simplifies to:

2x - 12 3x 8

Step 2: Rearrange the Equation

The next step is to arrange the equation so that all terms involving x are on one side, and the constant terms are on the other. Subtract 2x from both sides:

-12 3x - 2x 8

This simplifies to:

-12 x 8

Step 3: Isolate the Variable x

To find the value of x, we need to isolate it by subtracting 8 from both sides:

-12 - 8 x 8 - 8

This simplifies to:

-20 x

Final Solution

The solution to the equation 0.5x - 3 0.75x 2 is:

x -20

To verify the solution, substitute x -20 back into the original equation:

0.5(-20) - 3 0.75(-20) 2

-10 - 3 -15 2

-13 -13

This confirms that our solution is correct.

Conclusion

Solving equations with fractions and variables requires a systematic approach. By following these steps, you can easily solve similar equations. Always ensure to check your solution by substituting the value back into the original equation. Practicing with various examples will help deepen your understanding and improve your problem-solving skills in algebra.