Algebraic equations like 1.5(x + 4) – 3 = 4.5(x – 2) often present a challenge, especially when faced with multiple-choice answers like 3, 4, 5, or 9. Solving these equations requires a clear understanding of algebraic principles, such as the distributive property, isolating variables, and balancing both sides. In this comprehensive article, we’ll break down the problem step by step, find the correct value of x, and explain each algebraic step in detail. Whether you’re a student preparing for exams or simply someone who loves solving puzzles, this guide will help you solve the equation accurately and efficiently.
The Problem
The equation 1.5(x + 4) – 3 = 4.5(x – 2) may look complex at first glance, but solving it requires a methodical approach. We begin by identifying each part of the equation and understanding what needs to be solved. The goal is to find the value of x that makes the equation true.
The first step is to expand the equation using the distributive property. We need to multiply 1.5 by both x and 4, and similarly, multiply 4.5 by x and -2. This expansion helps us simplify the equation, making it easier to isolate the variable x.
Next, we rearrange the equation by moving like terms to one side. The objective here is to have all terms containing x on one side and the constant terms on the other. This rearrangement is crucial for solving the equation correctly. We must perform the same operation on both sides of the equation to maintain balance.
After isolating x, we can solve for its value by simplifying the equation. Finally, we verify our solution by plugging the value of x back into the original equation to ensure it satisfies the equation. This verification step is essential to confirm that we have found the correct answer.
Solving Step-by-Step for Clarity
To solve the equation 1.5(x + 4) – 3 = 4.5(x – 2), we need a clear step-by-step approach. Begin by distributing the coefficients: 1.5 multiplies both x and 4, while 4.5 multiplies both x and -2. This expands the equation to 1.5x + 6 – 3 = 4.5x – 9. Next, simplify by combining like terms: 1.5x + 3 = 4.5x – 9. Move the terms containing x to one side by subtracting 1.5x from both sides, giving us 3 = 3x – 9. Add 9 to both sides to isolate 3x, resulting in 12 = 3x. Finally, divide by 3 to solve for x, yielding x = 4. Verifying the solution by substituting x = 4 back into the original equation confirms that both sides are equal, ensuring the correctness of the answer.
Common Mistakes to Avoid
When solving 1.5(x + 4) – 3 = 4.5(x – 2), students often make errors that lead to incorrect solutions. One common mistake is failing to apply the distributive property correctly, which can result in missing terms. Another frequent error is improper handling of negative signs, particularly when expanding 4.5(x – 2). It’s crucial to simplify the equation properly by combining like terms. Additionally, forgetting to balance both sides of the equation when isolating x can lead to incorrect answers. Always verify the solution by substituting the value of x back into the original equation. This helps catch any errors made during the process. In this case, x = 4 is the correct solution.
Step-by-Step Solution with Headings
Distributing the Terms
To solve 1.5(x + 4) – 3 = 4.5(x – 2), start by distributing the coefficients:
- 1.5 * x = 1.5x
- 1.5 * 4 = 6
- 4.5 * x = 4.5x
- 4.5 * -2 = -9
This simplifies the equation to: 1.5x + 6 – 3 = 4.5x – 9
Simplifying the Equation
Now, simplify the equation by combining like terms:
- 1.5x + 3 = 4.5x – 9
Isolating x
Next, move the x terms to one side and the constant terms to the other:
- Subtract 1.5x from both sides: 3 = 3x – 9
Solving for x
Add 9 to both sides to isolate 3x:
- 12 = 3x
- Divide by 3: x = 4
Verification of the Solution
To verify, substitute x = 4 back into the original equation:
- 1.5(4 + 4) – 3 = 4.5(4 – 2)
- 12 – 3 = 9, which is true.
Thus, the correct value of x is 4.
Key Steps in Solving the Equation
- Distribute the coefficients: Apply the distributive property to expand the equation.
- Combine like terms: Simplify the equation by grouping similar terms.
- Move terms: Rearrange the equation to isolate the variable x.
- Solve for x: Perform arithmetic operations to find the value of x.
- Verify the solution: Plug the value of x back into the original equation to check correctness.
Analysis of Solving Linear Equations
Linear equations, such as 1.5(x + 4) – 3 = 4.5(x – 2), are fundamental in algebra. They represent equations where the variable is of the first degree, meaning it is not raised to any power other than one. Solving such equations requires a good grasp of algebraic properties and techniques.
The first step in solving any linear equation is to simplify both sides as much as possible. This involves distributing any coefficients, combining like terms, and isolating the variable. The key to solving linear equations is understanding that whatever operation you perform on one side of the equation must also be done to the other side to keep the equation balanced.
In this equation, the distributive property is used first, which is a rule that states you must multiply a term outside the parentheses by each term inside the parentheses. This expansion simplifies the equation, allowing us to group and move terms efficiently. By isolating the variable on one side, we can solve for its value. The final step is to check our solution by substituting it back into the original equation.
Understanding these steps is crucial, as linear equations form the basis for more complex algebraic problems. Mastering the process of solving linear equations can help solve more advanced mathematical problems.
Using the Keyword in Headings
What is the value of x in the equation 1.5(x + 4) – 3 = 4.5(x – 2)?
This equation is a classic example of a linear equation that can be solved using algebraic principles.
How to Isolate x in the Given Equation
To solve for x, start by distributing the coefficients and then rearrange the equation to isolate the variable.
Why is Verification Important?
Verification ensures that the value of x found is correct and satisfies the original equation.
Conclusion
In solving the equation 1.5(x + 4) – 3 = 4.5(x – 2), we used basic algebraic techniques like the distributive property, combining like terms, and isolating the variable. The value of x was determined to be 4, and verifying this solution confirmed its correctness. Understanding how to approach such equations helps build a strong foundation in algebra, making it easier to solve more complex problems in mathematics.
FAQ’s
Q. What is the first step in solving the equation 1.5(x + 4) – 3 = 4.5(x – 2)?
A. The first step is to apply the distributive property to expand the equation.
Q. What does it mean to isolate the variable in an equation?
A. Isolating the variable means moving all terms containing the variable to one side of the equation.
Q. What is the value of x in the equation 1.5(x + 4) – 3 = 4.5(x – 2)?
A. The value of x is 4.
Q. How do you verify the solution to an equation?
A. Substitute the solution back into the original equation to ensure both sides are equal.
Q. Why is it important to simplify equations?
A. Simplifying equations makes it easier to solve them and reduces the risk of errors.
