Solving The Algebraic Equation 2/3(x-1) - (x+2) = 1/4 A Step-by-Step Guide

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Hey guys! Today, we're diving into a classic algebra problem: solving the equation 2/3(x-1) - (x+2) = 1/4. Don't worry if it looks a bit intimidating at first. We'll break it down step-by-step, so you'll be a pro in no time. Solving algebraic equations is a fundamental skill in mathematics, and mastering it opens doors to more advanced concepts. This particular equation involves fractions, parentheses, and variables, making it a great exercise for honing your algebraic manipulation skills. By understanding the process of solving this equation, you'll gain confidence in your ability to tackle similar problems. The beauty of algebra lies in its systematic approach to problem-solving. Each step we take is guided by mathematical principles, ensuring that we arrive at the correct solution. This approach is not only applicable to solving equations but also to various other mathematical and real-world problems. So, let's embark on this journey of algebraic exploration and unravel the solution to this equation together!

1. Clearing the Fractions: Multiplying Both Sides

The first thing we want to do is get rid of those pesky fractions. Trust me, it'll make the equation much easier to work with. To do this, we need to find the least common multiple (LCM) of the denominators, which are 3 and 4. The LCM of 3 and 4 is 12. So, we'll multiply both sides of the equation by 12. This is a crucial step in simplifying the equation. By eliminating the fractions, we transform the equation into a more manageable form, paving the way for subsequent steps. Remember, whatever operation we perform on one side of the equation, we must also perform on the other side to maintain equality. This principle is the cornerstone of equation solving. Multiplying both sides by the LCM not only clears the fractions but also ensures that the equation remains balanced. This step highlights the importance of understanding and applying the fundamental properties of equality in algebra. By mastering this technique, you'll be well-equipped to tackle equations involving fractions with confidence. It's like having a secret weapon against those fractional foes! So, let's wield this weapon and conquer the fractions in our equation!

12∗[23(x−1)−(x+2)]=12∗[14]12 * [\frac{2}{3}(x-1) - (x+2)] = 12 * [\frac{1}{4}]

This gives us:

8(x−1)−12(x+2)=38(x-1) - 12(x+2) = 3

2. Distributing: Expanding the Parentheses

Now that we've cleared the fractions, let's tackle those parentheses. We'll use the distributive property, which means multiplying the number outside the parentheses by each term inside. Distribution is a fundamental algebraic technique that allows us to simplify expressions containing parentheses. It involves applying the distributive property of multiplication over addition or subtraction. In this step, we'll distribute the 8 and the -12 to the terms inside their respective parentheses. This process expands the expression and removes the parentheses, making it easier to combine like terms in the subsequent steps. Remember to pay close attention to the signs when distributing, as a misplaced sign can lead to an incorrect solution. The distributive property is a versatile tool that is widely used in various mathematical contexts, including algebra, calculus, and beyond. By mastering this technique, you'll gain a valuable skill that will serve you well in your mathematical journey. So, let's distribute those numbers and watch the parentheses disappear!

8x−8−12x−24=38x - 8 - 12x - 24 = 3

3. Combining Like Terms: Simplifying the Equation

Next up, we need to combine like terms. This means grouping together the terms with 'x' and the constant terms (the numbers without 'x'). Combining like terms is a crucial step in simplifying algebraic expressions and equations. It involves identifying terms that have the same variable raised to the same power and then adding or subtracting their coefficients. In this step, we'll combine the 'x' terms (8x and -12x) and the constant terms (-8 and -24). This process reduces the number of terms in the equation, making it easier to isolate the variable and solve for its value. Combining like terms not only simplifies the equation but also helps to reveal its underlying structure. By organizing the terms in a systematic way, we gain a clearer understanding of the relationships between the variables and constants. This technique is widely used in various mathematical contexts and is an essential skill for algebraic manipulation. So, let's gather our like terms and watch the equation become more streamlined!

(8x−12x)+(−8−24)=3(8x - 12x) + (-8 - 24) = 3

This simplifies to:

−4x−32=3-4x - 32 = 3

4. Isolating the Variable: Getting 'x' Alone

Our goal is to get 'x' all by itself on one side of the equation. To do this, we'll first add 32 to both sides. Isolating the variable is the ultimate goal in solving equations. It involves performing algebraic operations to get the variable term by itself on one side of the equation. In this step, we'll add 32 to both sides of the equation to eliminate the constant term on the left side. This operation moves us closer to isolating the 'x' term. Remember, whatever operation we perform on one side of the equation, we must also perform on the other side to maintain equality. This principle is the foundation of equation solving. Isolating the variable requires a systematic approach, involving a series of steps designed to undo the operations that are being performed on the variable. By mastering this technique, you'll gain the ability to solve a wide range of equations and unlock the secrets of algebra. So, let's isolate 'x' and unveil its value!

−4x−32+32=3+32-4x - 32 + 32 = 3 + 32

−4x=35-4x = 35

Now, we'll divide both sides by -4:

−4x−4=35−4\frac{-4x}{-4} = \frac{35}{-4}

5. The Solution: Finding the Value of 'x'

Finally, we have our solution! The value of 'x' is: Finding the solution is the culmination of all the steps we've taken. It's the moment when we unveil the value of the variable that satisfies the equation. In this step, we'll perform the final division to isolate 'x' and determine its value. The solution represents the point where the equation is balanced, where the left side is equal to the right side. Finding the solution is not just about getting the right answer; it's about understanding the process and the underlying principles of algebra. It's about developing the ability to think logically and systematically to solve problems. The solution is the reward for our efforts, the validation of our algebraic journey. So, let's celebrate our success and the power of mathematics!

x=−354x = -\frac{35}{4}

So, there you have it! The solution to the equation 2/3(x-1) - (x+2) = 1/4 is x = -35/4. Solving equations like this is a fundamental skill in algebra, and with practice, you'll become a whiz at it. Remember the key steps: clear fractions, distribute, combine like terms, and isolate the variable. Keep practicing, and you'll be solving even more complex equations in no time!