Solving 2x - 5 = 2(x + 9): Find The Solution Set

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Let's dive into solving the equation 2x - 5 = 2(x + 9). This type of problem is a staple in algebra, and understanding how to solve it will build a strong foundation for more complex math challenges. In this comprehensive guide, we'll walk through the steps, explain the logic, and explore why we arrive at a particular solution (or, in some cases, no solution at all!). We will explore each step to make sure it’s crystal clear and leave you confident in tackling similar equations.

Understanding the Basics of Linear Equations

Before we jump into the specific equation, it's helpful to understand the basics of linear equations. A linear equation is an equation where the highest power of the variable (in this case, 'x') is 1. These equations represent a straight line when graphed, hence the name “linear.” The main goal when solving a linear equation is to isolate the variable on one side of the equation. This means getting 'x' by itself so we can see what value it needs to be to make the equation true.

To isolate the variable, we use a few key principles:

  • The Golden Rule of Algebra: What you do to one side of the equation, you must do to the other. This ensures the equation remains balanced.
  • Inverse Operations: To undo an operation, we use its inverse. For example, the inverse of addition is subtraction, and the inverse of multiplication is division.
  • The Distributive Property: This property allows us to multiply a number across a sum or difference. For example, a(b + c) = ab + ac.

With these principles in mind, let's tackle our equation.

Step-by-Step Solution of 2x - 5 = 2(x + 9)

Our equation is 2x - 5 = 2(x + 9). Let's break down the solution process step by step:

Step 1: Apply the Distributive Property

The first thing we need to do is simplify the right side of the equation. We have 2 multiplied by the expression (x + 9). Using the distributive property, we multiply 2 by both terms inside the parentheses:

2 * (x + 9) = 2 * x + 2 * 9 = 2x + 18

Now our equation looks like this:

2x - 5 = 2x + 18

Step 2: Gather the 'x' Terms

Next, we want to get all the terms with 'x' on one side of the equation. To do this, we can subtract 2x from both sides. Remember, whatever we do to one side, we must do to the other to keep the equation balanced:

2x - 5 - 2x = 2x + 18 - 2x

This simplifies to:

-5 = 18

Step 3: Analyze the Result

Now, this is where things get interesting. We've simplified the equation to -5 = 18. This statement is clearly false. Negative five does not equal eighteen. So, what does this mean for our equation?

Interpreting the Outcome: No Solution

When solving an equation, if you arrive at a statement that is always false (like -5 = 18), it means that the original equation has no solution. There is no value of 'x' that will make the equation true.

Think about it this way: we were trying to find a value for 'x' that would balance the two sides of the equation. But after simplifying, we ended up with a contradiction. No matter what value we plug in for 'x', the left side will never equal the right side.

Why Does This Happen?

This outcome often occurs when we have parallel lines represented by the two sides of the equation. In simpler terms, the 'x' terms canceled out, leaving us with a false statement about the constants. This indicates that the lines never intersect, and therefore, there's no solution.

Expressing the Solution Set

Since there's no solution, we express the solution set as an empty set. This can be represented in a few ways:

  • { } (an empty set with curly braces)
  • ∅ (the null set symbol)

So, the answer to our original question is that the solution set is { } or ∅.

Common Mistakes to Avoid

When solving equations like this, it's easy to make small errors that lead to incorrect solutions. Here are a few common mistakes to watch out for:

  • Distributive Property Errors: Make sure you multiply the number outside the parentheses by every term inside the parentheses.
  • Sign Errors: Pay close attention to the signs (+ and -) when adding, subtracting, multiplying, and dividing.
  • Incorrectly Combining Like Terms: Only combine terms that have the same variable and exponent.
  • Misinterpreting the Result: If you get a false statement (like -5 = 18), don't try to force a solution. Recognize that it means there's no solution.

Practice Problems for Mastering Equation Solving

The best way to solidify your understanding of solving equations is through practice. Here are a few similar problems you can try:

  1. 3(x - 2) = 3x + 5
  2. 4x + 1 = 4(x - 2)
  3. -2(x + 1) = -2x + 3

Work through these problems step-by-step, and remember to check your answers. If you get stuck, review the steps we outlined earlier in this guide.

Conclusion

Solving the equation 2x - 5 = 2(x + 9) ultimately leads us to the conclusion that there is no solution. This occurs because the equation simplifies to a false statement, indicating that no value of 'x' can satisfy the equation. By understanding the basic principles of linear equations and practicing regularly, you can confidently tackle these types of problems. Remember to apply the distributive property, combine like terms, and carefully interpret the results.

Mastering these skills will not only help you in algebra but also in various real-world problem-solving scenarios. So, keep practicing, and don't hesitate to revisit these concepts whenever you need a refresher.

For additional resources and practice problems, consider visiting websites like Khan Academy's Algebra Section, which offers a wealth of materials on linear equations and other algebraic topics.