Solving F(x) = -12 For The Absolute Value Function F(x) = -5|x+1| + 3
Introduction
In this comprehensive article, we delve into the realm of absolute value functions and tackle the problem of finding the specific values of x for which the function f(x) = -5|x + 1| + 3 equals -12. This is a common type of problem in algebra and precalculus, and understanding how to solve it is crucial for mastering these mathematical concepts. We will guide you through a step-by-step solution, explaining the underlying principles and techniques involved. By the end of this article, you will not only be able to solve this particular problem but also gain a deeper understanding of absolute value equations and how to approach them. Absolute value functions may seem intimidating at first glance, but with a systematic approach, they can be easily conquered. So, let's embark on this mathematical journey and unravel the intricacies of this equation.
Understanding Absolute Value
Before diving into the specifics of the problem, it's crucial to have a solid grasp of the concept of absolute value. The absolute value of a number is its distance from zero on the number line, regardless of direction. It's denoted by two vertical bars surrounding the number, like this: |x|. For instance, |3| = 3 because 3 is 3 units away from zero, and |-3| = 3 because -3 is also 3 units away from zero. This property of absolute value is fundamental to solving equations involving absolute value functions. Essentially, the absolute value function transforms any negative input into its positive counterpart while leaving positive inputs unchanged. This creates a piecewise nature to absolute value functions, which is critical to consider when solving equations. The absolute value function can be formally defined as follows:
|x| = x, if x ≥ 0 |x| = -x, if x < 0
This definition highlights the two possible scenarios we need to consider when dealing with absolute value equations: the expression inside the absolute value bars is either non-negative or negative. Understanding this dual nature is the key to correctly solving these types of equations. Now that we have a firm understanding of absolute value, let's proceed to the problem at hand.
Setting up the Equation
Our objective is to find the values of x that satisfy the equation f(x) = -12, where f(x) = -5|x + 1| + 3. To begin, we substitute -12 for f(x) in the equation, giving us: -12 = -5|x + 1| + 3. This equation now presents a clear path forward. Our next step involves isolating the absolute value term, which will allow us to apply the properties of absolute value and solve for x. This process involves performing algebraic manipulations to get the absolute value expression by itself on one side of the equation. Once the absolute value term is isolated, we can then consider the two possible cases that arise from the definition of absolute value. This strategic approach breaks down the problem into manageable steps, making it easier to solve. By isolating the absolute value term, we set the stage for the next crucial phase of the solution process.
Isolating the Absolute Value
To isolate the absolute value term |x + 1|, we need to perform a few algebraic steps. First, we subtract 3 from both sides of the equation: -12 - 3 = -5|x + 1| + 3 - 3, which simplifies to -15 = -5|x + 1|. Next, we divide both sides by -5 to completely isolate the absolute value: -15 / -5 = -5|x + 1| / -5, resulting in 3 = |x + 1|. Now, the equation is in a form where we can directly apply the definition of absolute value. The equation 3 = |x + 1| tells us that the expression (x + 1) is either 3 units away from zero in the positive direction or 3 units away from zero in the negative direction. This is the crucial step where we transition from an equation with an absolute value to two separate linear equations, each representing one of these possibilities. By successfully isolating the absolute value term, we have paved the way for solving for x in each case.
Solving for x: Two Cases
Now that we have the equation 3 = |x + 1|, we can consider the two possible cases that arise from the definition of absolute value:
Case 1: The expression inside the absolute value is positive or zero. In this case, |x + 1| = x + 1, so our equation becomes 3 = x + 1. To solve for x, we subtract 1 from both sides: 3 - 1 = x + 1 - 1, which gives us x = 2. This is one potential solution to our original equation.
Case 2: The expression inside the absolute value is negative. In this case, |x + 1| = -(x + 1), so our equation becomes 3 = -(x + 1). To solve for x, we first distribute the negative sign: 3 = -x - 1. Then, we add 1 to both sides: 3 + 1 = -x - 1 + 1, which gives us 4 = -x. Finally, we multiply both sides by -1 to solve for x: -4 = x. This gives us our second potential solution.
These two cases cover all possibilities for the absolute value, ensuring that we find all values of x that satisfy the original equation. We now have two potential solutions: x = 2 and x = -4. To ensure the accuracy of our solutions, we must verify them by substituting them back into the original equation.
Verifying the Solutions
To ensure that our solutions are correct, we need to substitute each value of x back into the original equation, f(x) = -5|x + 1| + 3, and check if it equals -12.
For x = 2:
f(2) = -5|2 + 1| + 3 f(2) = -5|3| + 3 f(2) = -5(3) + 3 f(2) = -15 + 3 f(2) = -12
Since f(2) = -12, the solution x = 2 is valid.
For x = -4:
f(-4) = -5|-4 + 1| + 3 f(-4) = -5|-3| + 3 f(-4) = -5(3) + 3 f(-4) = -15 + 3 f(-4) = -12
Since f(-4) = -12, the solution x = -4 is also valid.
Both solutions, x = 2 and x = -4, satisfy the original equation. This verification step is crucial in solving absolute value equations because it confirms that our solutions are not extraneous, which can sometimes occur due to the nature of absolute value. By verifying our solutions, we can confidently conclude that we have found the correct values of x.
Conclusion
In conclusion, the values of x for which f(x) = -5|x + 1| + 3 = -12 are x = 2 and x = -4. We arrived at these solutions by carefully applying the definition of absolute value, isolating the absolute value term, and considering both positive and negative cases. We then verified our solutions to ensure their accuracy. This process highlights the importance of a systematic approach when solving absolute value equations. Understanding the concept of absolute value, breaking down the problem into cases, and verifying solutions are all essential steps in mastering these types of problems. We hope this comprehensive guide has provided you with a clear understanding of how to solve absolute value equations and has equipped you with the necessary skills to tackle similar problems in the future. Remember, practice is key to mastering mathematical concepts, so keep practicing and exploring the fascinating world of algebra!
