Solving Absolute Value Equations A Step-by-Step Guide To |6u - 18| = 12

Absolute value equations might seem daunting at first, but with a systematic approach, they can be easily conquered. The key to solving these equations lies in understanding the nature of absolute value, which represents the distance of a number from zero on the number line. This means that an absolute value expression can have two possible solutions: a positive and a negative value. In this comprehensive guide, we will delve into the intricacies of solving the absolute value equation $|6u - 18| = 12$, providing a step-by-step solution and exploring the underlying concepts.

Understanding Absolute Value

Before we dive into the solution, let's clarify the concept of absolute value. The absolute value of a number, denoted by $|x|$, is its distance from zero on the number line. This distance is always non-negative, meaning it is either positive or zero. For instance, the absolute value of 5, written as $|5|$, is 5 because 5 is 5 units away from zero. Similarly, the absolute value of -5, written as $|-5|$, is also 5 because -5 is also 5 units away from zero. This fundamental property of absolute value is crucial for solving absolute value equations.

The Two Cases

When dealing with an absolute value equation like $|6u - 18| = 12$, we need to consider two distinct cases: the expression inside the absolute value can be either positive or negative, but its distance from zero is always 12. This leads to two separate equations that we need to solve:

1. Case 1: The expression inside the absolute value is positive or zero: $6u - 18 = 12$
2. Case 2: The expression inside the absolute value is negative: $6u - 18 = -12$

By considering both cases, we ensure that we capture all possible solutions to the original absolute value equation.

Step-by-Step Solution for $|6u - 18| = 12$

Now, let's solve the equation $|6u - 18| = 12$ step-by-step, considering both cases:

Case 1: $6u - 18 = 12$

1. Isolate the term with the variable: To isolate the term with the variable, which is $6u$, we need to eliminate the constant term, -18, from the left side of the equation. We can do this by adding 18 to both sides of the equation: $6u - 18 + 18 = 12 + 18$ This simplifies to: $6u = 30$
2. Solve for u: Now that we have isolated the term with the variable, we can solve for u by dividing both sides of the equation by the coefficient of u, which is 6: rac{6u}{6} = rac{30}{6} This simplifies to: $u = 5$ So, one possible solution is $u = 5$.

Case 2: $6u - 18 = -12$

1. Isolate the term with the variable: Similar to Case 1, we need to isolate the term with the variable, $6u$. We add 18 to both sides of the equation: $6u - 18 + 18 = -12 + 18$ This simplifies to: $6u = 6$
2. Solve for u: To solve for u, we divide both sides of the equation by 6: rac{6u}{6} = rac{6}{6} This simplifies to: $u = 1$ Therefore, the second possible solution is $u = 1$.

The Solutions

We have found two possible solutions for the equation $|6u - 18| = 12$: $u = 5$ and $u = 1$. To ensure accuracy, it's always a good practice to check these solutions by substituting them back into the original equation.

Checking the Solutions

1. Check u = 5: Substitute $u = 5$ into the original equation: $|6(5) - 18| = 12$ $|30 - 18| = 12$ $|12| = 12$ $12 = 12$ (This is true) So, $u = 5$ is a valid solution.
2. Check u = 1: Substitute $u = 1$ into the original equation: $|6(1) - 18| = 12$ $|6 - 18| = 12$ $|-12| = 12$ $12 = 12$ (This is true) Thus, $u = 1$ is also a valid solution.

Both solutions, $u = 5$ and $u = 1$, satisfy the original equation. Therefore, the solutions to the equation $|6u - 18| = 12$ are $u = 1, 5$.

Key Concepts and Takeaways

Let's recap the key concepts and takeaways from solving this absolute value equation:

• Absolute Value: The absolute value of a number is its distance from zero on the number line, and it is always non-negative.
• Two Cases: When solving absolute value equations of the form $|ax + b| = c$, we need to consider two cases: $ax + b = c$ and $ax + b = -c$.
• Isolate the Absolute Value: Before splitting the equation into two cases, make sure to isolate the absolute value expression on one side of the equation.
• Solve Each Case: Solve each case separately by applying standard algebraic techniques, such as adding or subtracting constants and dividing by coefficients.
• Check Solutions: After finding the possible solutions, always check them by substituting them back into the original equation to ensure they are valid.

Additional Tips for Solving Absolute Value Equations

Here are some additional tips to help you master solving absolute value equations:

• Simplify First: If the equation contains any like terms or expressions that can be simplified, do so before proceeding with the two cases.
• No Solution: Be aware that some absolute value equations may have no solution. This occurs when the absolute value expression is equal to a negative number, which is impossible since absolute values are always non-negative. For example, the equation $|x + 2| = -3$ has no solution.
• Extraneous Solutions: It's possible to obtain extraneous solutions when solving absolute value equations, which are solutions that satisfy the transformed equations but not the original equation. This is why checking solutions is crucial.
• Graphical Interpretation: Absolute value equations can also be interpreted graphically. The solutions represent the points where the graph of the absolute value function intersects the horizontal line representing the constant value on the right side of the equation.

Practice Problems

To solidify your understanding of solving absolute value equations, try solving the following practice problems:

1. $|2x - 1| = 5$
2. $|3u + 6| = 9$
3. $|4v - 8| = 0$
4. $|5w + 10| = -2$ (Hint: Think about the possibility of no solution)

By working through these practice problems, you'll gain confidence and proficiency in solving absolute value equations.

Conclusion

Solving absolute value equations involves considering two cases and applying standard algebraic techniques. By understanding the concept of absolute value, following a systematic approach, and checking your solutions, you can confidently solve a wide range of absolute value equations. Remember to practice regularly and apply the tips and techniques discussed in this comprehensive guide to excel in your mathematical endeavors.