Solving Absolute Value Equations A Step-by-Step Guide To |3y + 9| = 3
Absolute value equations can seem daunting at first, but with a systematic approach, they become quite manageable. In this article, we will delve into the process of solving the equation |3y + 9| = 3, providing a clear, step-by-step guide that will equip you with the skills to tackle similar problems with confidence. Understanding absolute value is crucial, as it represents the distance of a number from zero, regardless of direction. This concept is key to solving equations involving absolute values, as it necessitates considering both positive and negative possibilities. We will explore the fundamental principles behind absolute value, and how they apply to solving equations. From setting up the initial equations to isolating the variable and verifying the solutions, each step will be elucidated with meticulous detail. By the end of this guide, you will not only be able to solve the given equation but also grasp the underlying concepts, making you proficient in handling a wide range of absolute value problems.
Understanding Absolute Value
Before diving into the equation, let's solidify our understanding of absolute value. The absolute value of a number is its distance from zero on the number line. This distance is always non-negative. For example, the absolute value of 5, denoted as |5|, is 5, and the absolute value of -5, denoted as |-5|, is also 5. This is because both 5 and -5 are 5 units away from zero. This property is crucial when dealing with equations involving absolute values. When we see an equation like |x| = a, it means that the expression inside the absolute value bars, x, can be either a or -a, since both of these values have a distance of 'a' from zero. This is the core concept that allows us to break down absolute value equations into two separate equations, which we can then solve individually. Grasping this principle is the first step in mastering the art of solving absolute value equations. The absolute value function essentially transforms any negative value into its positive counterpart, while leaving positive values unchanged. This behavior stems directly from the definition of absolute value as a distance, which cannot be negative. Therefore, when we encounter |3y + 9| = 3, we need to consider both scenarios: when the expression inside the absolute value, 3y + 9, is equal to 3, and when it is equal to -3. This dual consideration is what distinguishes solving absolute value equations from solving regular algebraic equations.
Setting Up the Equations
Now that we understand absolute value, let's apply this knowledge to our equation, |3y + 9| = 3. Since the absolute value of an expression is its distance from zero, the expression inside the absolute value bars, 3y + 9, can be either 3 or -3. This gives us two separate equations to solve:
- 3y + 9 = 3
- 3y + 9 = -3
This is the crucial step in solving absolute value equations. By recognizing that the expression inside the absolute value bars can be both positive and negative, we effectively transform one absolute value equation into two linear equations. This allows us to use familiar algebraic techniques to isolate the variable and find the solutions. The key is to remember this bifurcation: always create two equations, one where the expression inside the absolute value equals the positive value on the other side of the equation, and another where it equals the negative value. Once these two equations are set up correctly, the rest of the solution process involves standard algebraic manipulations. This step is not just a procedural trick; it is a direct consequence of the definition of absolute value. The absolute value function maps both a number and its negative counterpart to the same positive value. Therefore, if |3y + 9| equals 3, it logically follows that 3y + 9 could have been either 3 or -3 before the absolute value was applied. Accurately setting up these two equations is paramount, as any error at this stage will propagate through the rest of the solution, leading to incorrect results. It's a fundamental branching point in the solution process, dictating the subsequent steps and ultimately determining the final answers.
Solving the First Equation: 3y + 9 = 3
Let's solve the first equation, 3y + 9 = 3. Our goal is to isolate the variable 'y'. To do this, we first subtract 9 from both sides of the equation:
3y + 9 - 9 = 3 - 9
This simplifies to:
3y = -6
Next, we divide both sides by 3:
3y / 3 = -6 / 3
This gives us our first solution:
y = -2
This process of isolating the variable is a cornerstone of algebra. Each step is designed to peel away the layers surrounding 'y' until it stands alone on one side of the equation. Subtracting 9 from both sides maintains the equality by performing the same operation on both sides of the equation. Similarly, dividing both sides by 3 isolates 'y' without disrupting the balance. This systematic approach is crucial for solving linear equations efficiently and accurately. The solution y = -2 represents a value that, when substituted back into the original equation, will make the equation true. However, this is only one potential solution, as we also need to consider the second equation derived from the absolute value. It is important to remember that solving one equation derived from an absolute value expression does not complete the task; the other possibility must also be explored. The rigor of algebra lies in this meticulous approach, ensuring that all potential solutions are identified and considered. The steps taken here are not merely mechanical; they are rooted in the fundamental properties of equality and the desire to isolate the unknown variable in a logical and consistent manner.
Solving the Second Equation: 3y + 9 = -3
Now, let's solve the second equation, 3y + 9 = -3. Again, we aim to isolate 'y'. We begin by subtracting 9 from both sides:
3y + 9 - 9 = -3 - 9
This simplifies to:
3y = -12
Next, we divide both sides by 3:
3y / 3 = -12 / 3
This gives us our second solution:
y = -4
The approach used here mirrors the process in solving the first equation, emphasizing the consistency and predictability of algebraic techniques. Subtracting 9 from both sides and then dividing by 3 are standard operations for isolating a variable in a linear equation. The solution y = -4 represents another value that, when plugged back into the original absolute value equation, will satisfy the equality. This highlights the key characteristic of absolute value equations: they often possess two distinct solutions. These solutions correspond to the two scenarios arising from the nature of absolute value – the expression inside the absolute value bars can be either positive or negative, leading to different values for the variable. Failing to consider both equations would result in an incomplete solution, missing one of the possible values for 'y'. The careful and methodical application of algebraic principles ensures that all potential solutions are identified. It is a testament to the power of algebra in providing a structured framework for solving equations, regardless of their complexity. This second solution, y = -4, complements the first solution, y = -2, and together they form the complete solution set for the original absolute value equation.
Verifying the Solutions
It's always a good practice to verify our solutions by substituting them back into the original equation, |3y + 9| = 3. Let's start with y = -2:
|3(-2) + 9| = | -6 + 9 | = | 3 | = 3
This solution checks out. Now, let's verify y = -4:
|3(-4) + 9| = | -12 + 9 | = | -3 | = 3
This solution also checks out. Therefore, our solutions y = -2 and y = -4 are correct.
Verification is a crucial step in the problem-solving process, especially with absolute value equations. It acts as a safeguard against potential errors made during the algebraic manipulations. By substituting the obtained values back into the original equation, we confirm whether they indeed satisfy the equality. This process not only validates the solutions but also deepens our understanding of the equation itself. In the case of y = -2, the substitution leads to |3|, which equals 3, confirming that -2 is a valid solution. Similarly, substituting y = -4 results in |-3|, which also equals 3, further solidifying the correctness of our solution. This step-by-step verification process leaves no room for doubt, ensuring that the solutions are accurate and consistent with the original problem. It is a hallmark of careful mathematical practice, reinforcing the importance of precision and attention to detail. The act of verification transforms the solution from a potential answer to a confirmed result, adding a layer of confidence and completeness to the problem-solving endeavor. Moreover, it reinforces the concept of absolute value, highlighting how both positive and negative expressions within the absolute value bars can lead to the same result.
Final Answer
Therefore, the solutions to the equation |3y + 9| = 3 are y = -2 and y = -4.
In conclusion, solving absolute value equations involves understanding the concept of absolute value, setting up two separate equations, solving each equation individually, and verifying the solutions. By following these steps, you can confidently solve a wide range of absolute value problems. The key lies in recognizing the dual nature of absolute value – the expression inside the absolute value bars can be either positive or negative, leading to two distinct possibilities that must be explored. This methodical approach, combined with a solid understanding of algebraic principles, will empower you to tackle these equations with ease and accuracy. The solutions y = -2 and y = -4 represent the complete set of values that satisfy the original equation, and their verification underscores the importance of checking one's work in mathematics. The process of solving |3y + 9| = 3 serves as a microcosm of the broader mathematical landscape, where careful reasoning, systematic execution, and a commitment to accuracy are paramount. By mastering the techniques presented here, you will not only be able to solve this specific equation but also develop a deeper appreciation for the elegance and power of algebra.
Therefore, the final answer is: y = -2, -4
