Solving Absolute Value Equations A Step-by-Step Guide To |x/6 - 7/12| = 11/6
Hey guys! Today, we're diving deep into solving an absolute value equation. These types of problems might seem tricky at first, but with a step-by-step approach, they become super manageable. We're going to break down the equation |x/6 - 7/12| = 11/6, exploring both the positive and negative scenarios. So, grab your pencils, and let's get started!
Understanding Absolute Value
Before we jump into the nitty-gritty, let's quickly recap what absolute value means. The absolute value of a number is its distance from zero on the number line. This means it's always non-negative. Think of it like this: |5| is 5, and |-5| is also 5. The absolute value strips away the sign, giving you the magnitude. When dealing with absolute value equations, this concept is crucial because it means we have to consider two possibilities: the expression inside the absolute value bars could be positive or negative.
The Core Concept
In essence, when you encounter an absolute value equation like |x/6 - 7/12| = 11/6, it's like saying, "The distance between the expression (x/6 - 7/12) and zero is 11/6." This leads us to two distinct equations that we need to solve separately. One equation assumes the expression inside the absolute value is equal to the positive value (11/6), and the other assumes it's equal to the negative value (-11/6). This split is the heart of solving absolute value equations, and mastering it will make these problems a breeze. Remember, it’s all about accounting for both directions on the number line – positive and negative – that result in the same distance from zero.
Visualizing Absolute Value
To really nail this concept, think about a number line. Imagine you're standing at zero, and you take steps either to the right (positive direction) or to the left (negative direction). The absolute value is simply the number of steps you've taken, regardless of the direction. So, if you take 5 steps to the right or 5 steps to the left, you've traveled a distance of 5 units. This visualization helps in understanding why we need to consider both positive and negative cases when solving absolute value equations. It's not just about the number; it's about the distance from zero, which can be achieved in two ways.
Breaking Down the Equation |x/6 - 7/12| = 11/6
Now, let's get our hands dirty with the equation at hand: |x/6 - 7/12| = 11/6. As we discussed, the absolute value means we have two scenarios to explore. This is the key to cracking the problem. We're essentially splitting this one equation into two separate equations, each representing a different possibility for the value inside the absolute value bars.
The Positive Scenario
First up, let's consider the scenario where the expression inside the absolute value is positive. This means that (x/6 - 7/12) is equal to 11/6. So, we can write our first equation as:
x/6 - 7/12 = 11/6
This equation represents the case where the distance from zero is achieved by moving in the positive direction. We're saying that the value of (x/6 - 7/12) itself is 11/6. This is a straightforward algebraic equation, and we'll solve it in the next section. But for now, just remember that this is one piece of the puzzle. We've accounted for the positive possibility, but we still need to consider the negative one.
The Negative Scenario
Next, we need to think about the possibility where the expression inside the absolute value is negative. This is where things might seem a bit tricky, but it's crucial for a complete solution. If the expression (x/6 - 7/12) is negative, then its absolute value will be the opposite of the expression. In other words, if (x/6 - 7/12) is -11/6, then |x/6 - 7/12| will still be 11/6. So, our second equation is:
x/6 - 7/12 = -11/6
This equation captures the scenario where the distance from zero is achieved by moving in the negative direction. It's equally important as the positive scenario, and we need to solve it to find the other possible value of x. By considering both positive and negative cases, we ensure that we've covered all the bases and found all possible solutions to the original absolute value equation. Remember, absolute value is all about distance from zero, and distance doesn't care about direction!
Solving the Negative Direction: x/6 - 7/12 = -11/6
Alright, let's tackle the negative scenario first. We've got the equation x/6 - 7/12 = -11/6. Our mission is to isolate x and find its value. Remember, guys, the key to solving any algebraic equation is to perform the same operations on both sides, keeping the equation balanced.
Step 1: Isolating the x term
Our first goal is to get the term with x (which is x/6) by itself on one side of the equation. To do this, we need to get rid of the -7/12. The opposite of subtracting 7/12 is adding 7/12, so let's add 7/12 to both sides of the equation:
x/6 - 7/12 + 7/12 = -11/6 + 7/12
On the left side, the -7/12 and +7/12 cancel each other out, leaving us with just x/6. Now, let's simplify the right side. We need to add -11/6 and 7/12. To do this, we need a common denominator. The least common multiple of 6 and 12 is 12, so let's convert -11/6 to an equivalent fraction with a denominator of 12. We can do this by multiplying both the numerator and denominator by 2:
-11/6 * 2/2 = -22/12
Now we can rewrite our equation as:
x/6 = -22/12 + 7/12
Step 2: Combining the Fractions
Now that we have a common denominator, we can easily add the fractions on the right side:
x/6 = (-22 + 7)/12
x/6 = -15/12
Step 3: Solving for x
We're almost there! Now we have x/6 = -15/12. To isolate x, we need to get rid of the division by 6. The opposite of dividing by 6 is multiplying by 6, so let's multiply both sides of the equation by 6:
(x/6) * 6 = (-15/12) * 6
On the left side, the multiplication and division by 6 cancel each other out, leaving us with just x. On the right side, we can simplify the multiplication:
x = (-15 * 6) / 12
x = -90/12
Step 4: Simplifying the Fraction
We've found a value for x, but it's always good practice to simplify fractions. Both -90 and 12 are divisible by 6, so let's divide both the numerator and denominator by 6:
x = (-90 / 6) / (12 / 6)
x = -15/2
So, our first solution is x = -15/2. That wasn't so bad, was it? We systematically isolated x, and now we have one of the solutions to our absolute value equation. But remember, we still have the positive scenario to tackle!
Tackling the Positive Direction
Okay, team, let's switch gears and focus on the positive scenario of our absolute value equation. This time, we're working with the equation x/6 - 7/12 = 11/6. Just like before, our ultimate goal is to isolate x and uncover its value. We'll follow a similar step-by-step approach as we did with the negative scenario, ensuring we keep the equation balanced along the way.
Isolating the x Term
First things first, we need to get the x/6 term all by its lonesome on one side of the equation. To achieve this, we'll counteract the -7/12 by adding 7/12 to both sides. This keeps our equation nice and balanced, just like a tightrope walker!
x/6 - 7/12 + 7/12 = 11/6 + 7/12
The -7/12 and +7/12 on the left side bid each other farewell, leaving us with x/6. Now, let's tidy up the right side. We need to combine 11/6 and 7/12. Remember, to add fractions, they need to have the same denominator. The smallest common denominator for 6 and 12 is 12, so let's transform 11/6 into an equivalent fraction with a denominator of 12. We can do this by multiplying both the numerator and the denominator by 2:
11/6 * 2/2 = 22/12
Now our equation looks a little sleeker:
x/6 = 22/12 + 7/12
Combining Those Fractions
Now that we've got a common denominator, adding the fractions on the right side is a piece of cake:
x/6 = (22 + 7) / 12
x/6 = 29/12
The Final Stretch Solving for x
We're on the home stretch! We've got x/6 = 29/12. To finally get x by itself, we need to undo the division by 6. The trusty opposite operation is multiplication, so we'll multiply both sides of the equation by 6:
(x/6) * 6 = (29/12) * 6
The multiplication and division by 6 on the left side cancel each other out, leaving us with our prize, x! On the right side, let's simplify the multiplication:
x = (29 * 6) / 12
x = 174 / 12
Simplifying to the Max
We've found a value for x, but let's make it shine by simplifying the fraction. Both 174 and 12 have a common factor of 6, so we'll divide both the numerator and the denominator by 6:
x = (174 / 6) / (12 / 6)
x = 29/2
And there you have it! Our second solution is x = 29/2. High fives all around for conquering the positive scenario!
Putting It All Together The Solutions
Awesome work, guys! We've successfully navigated both the negative and positive scenarios of the absolute value equation |x/6 - 7/12| = 11/6. That means we've uncovered both possible values for x that make the equation true.
Reviewing Our Findings
Let's take a moment to recap our journey. We started with the absolute value equation, recognized the two possibilities (positive and negative), and split the problem into two separate equations. We meticulously solved each equation, keeping everything balanced and simplified. Now, we have our two solutions:
- For the negative scenario, we found x = -15/2.
- For the positive scenario, we found x = 29/2.
These are the two values of x that, when plugged back into the original equation, will make the absolute value equal to 11/6. It's like we've found the hidden keys that unlock the equation!
The Complete Solution Set
So, the complete solution set for the equation |x/6 - 7/12| = 11/6 is {-15/2, 29/2}. This means that if you were to graph this equation, these two values would be the points where the graph intersects a certain line. But for now, just focus on the algebraic solution we've achieved. You've tackled an absolute value equation head-on, and that's a fantastic accomplishment!
Key Takeaways for Solving Absolute Value Equations
Before we wrap things up, let's distill the essence of what we've learned into some key takeaways. These principles will serve you well as you encounter more absolute value equations in the future. Mastering these concepts will make you a true absolute value equation-solving pro!
Always Consider Both Scenarios
The most crucial thing to remember is that absolute value equations inherently have two possibilities: the expression inside the absolute value bars can be either positive or negative. This is because absolute value measures distance from zero, which can be achieved in two directions. Never forget to split the original equation into two separate equations, one for the positive scenario and one for the negative scenario. This is the cornerstone of solving these types of problems.
Isolate the Absolute Value First
Before you split the equation into two scenarios, make sure you've isolated the absolute value expression on one side of the equation. This means that you should get the absolute value bars all by themselves before you start dealing with the positive and negative cases. If there are any terms added, subtracted, multiplied, or divided outside the absolute value, take care of those first. This will make your subsequent steps much cleaner and easier to manage.
Solve Each Equation Carefully
Once you've split the equation into two scenarios, you'll have two separate algebraic equations to solve. Treat each of these equations as its own individual problem. Use the same techniques you would use for any algebraic equation: perform the same operations on both sides, simplify as much as possible, and isolate the variable. Pay close attention to signs and fractions, and double-check your work to minimize errors. Accuracy is key to getting the correct solutions.
Check Your Solutions
As a final step, it's always a good idea to check your solutions by plugging them back into the original absolute value equation. This will ensure that your solutions are valid and that you haven't made any mistakes along the way. If a solution doesn't work when you plug it back in, it's called an extraneous solution, and you should discard it. Checking your solutions is a crucial step in ensuring the correctness of your answers.
Practice Makes Perfect More Examples
To truly solidify your understanding of solving absolute value equations, practice is essential. The more you work through different examples, the more comfortable and confident you'll become. So, let's explore a few more examples to hone your skills.
Example 1 |2x + 3| = 7
First, we split this into two equations:
- 2x + 3 = 7
- 2x + 3 = -7
Solving the first equation:
- 2x = 4
- x = 2
Solving the second equation:
- 2x = -10
- x = -5
So, the solutions are x = 2 and x = -5.
Example 2 |x/3 - 1| = 4
Splitting into two equations:
- x/3 - 1 = 4
- x/3 - 1 = -4
Solving the first equation:
- x/3 = 5
- x = 15
Solving the second equation:
- x/3 = -3
- x = -9
Thus, the solutions are x = 15 and x = -9.
Example 3 |4 - x| = 9
Splitting into two equations:
- 4 - x = 9
- 4 - x = -9
Solving the first equation:
- -x = 5
- x = -5
Solving the second equation:
- -x = -13
- x = 13
Therefore, the solutions are x = -5 and x = 13.
By working through these examples, you've gained even more experience in applying the techniques for solving absolute value equations. Remember the key takeaways, and keep practicing! With each problem you solve, you'll become more adept at handling these equations.
Conclusion
And there we have it! You've successfully learned how to solve the absolute value equation |x/6 - 7/12| = 11/6, and you've gained a solid understanding of the core concepts behind absolute value equations. Remember to always consider both positive and negative scenarios, isolate the absolute value expression, and solve each equation carefully. With practice, you'll become a master of these types of problems. Keep up the great work, and happy solving!