Solving The Absolute Value Equation $-4|-2x+6|=-24$ A Step-by-Step Guide

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Absolute value equations can seem daunting at first, but with a systematic approach, they become quite manageable. This article will delve into the step-by-step solution of the equation $-4|-2x+6|=-24$, providing a comprehensive understanding of the underlying concepts and techniques. We will not only solve this specific equation but also equip you with the knowledge to tackle similar problems confidently. Understanding absolute value equations is crucial for various mathematical applications, and this guide aims to make the process clear and accessible.

Understanding Absolute Value

Before we dive into solving the equation, it's essential to understand the concept of absolute value. The absolute value of a number is its distance from zero on the number line. This distance is always non-negative. Mathematically, the absolute value of a number $x$, denoted as |x|, is defined as:

  • |x| = x, if x ≥ 0
  • |x| = -x, if x < 0

For example, |5| = 5 and |-5| = -(-5) = 5. The absolute value essentially strips away the sign of the number, leaving only its magnitude. This concept is fundamental when solving equations involving absolute values because it introduces the possibility of two different scenarios, one where the expression inside the absolute value is positive or zero, and another where it is negative. When dealing with absolute value equations, this understanding allows us to split the problem into manageable cases, ensuring that we account for all potential solutions. Recognizing that absolute value represents distance is key to correctly interpreting and solving these types of equations. Therefore, it’s important to remember that the expression inside the absolute value bars can be either positive or negative, leading to different solutions.

Step-by-Step Solution of $-4|-2x+6|=-24$

Now, let's tackle the equation $-4|-2x+6|=-24$. We will break down the solution into several clear steps to ensure a thorough understanding.

Step 1: Isolate the Absolute Value Expression

The first step in solving any absolute value equation is to isolate the absolute value expression. In our case, we need to isolate |-2x+6|. To do this, we divide both sides of the equation by -4:

−4∣−2x+6∣−4=−24−4{ \frac{-4|-2x+6|}{-4} = \frac{-24}{-4} }

This simplifies to:

∣−2x+6∣=6{ |-2x+6| = 6 }

Isolating the absolute value is crucial because it sets the stage for the next steps, where we consider the two possible cases. By isolating the absolute value, we make it clear that the expression inside the absolute value bars, in this case, -2x+6, can be either 6 or -6. This isolation is a standard technique when solving absolute value equations and helps in simplifying the problem into more manageable parts. Correctly performing this step is essential for arriving at the correct solutions. Therefore, always ensure that the absolute value expression is isolated before proceeding further.

Step 2: Set Up Two Equations

Since the absolute value of an expression can be either positive or negative, we need to consider both possibilities. This means we will set up two separate equations:

Case 1: The expression inside the absolute value is equal to 6:

−2x+6=6{ -2x+6 = 6 }

Case 2: The expression inside the absolute value is equal to -6:

−2x+6=−6{ -2x+6 = -6 }

This step is the heart of solving absolute value equations. By considering both the positive and negative possibilities of the expression inside the absolute value, we ensure that we capture all potential solutions. It's crucial to understand that the absolute value equation splits into these two cases due to the definition of absolute value itself. Each case represents a different scenario, and solving both is necessary to obtain a complete solution. This bifurcation is a defining characteristic of absolute value problems, and mastering this step is key to successfully solving them. Thus, always remember to create two equations when tackling an absolute value problem.

Step 3: Solve Each Equation

Now, we solve each equation separately.

Case 1: -2x+6 = 6

Subtract 6 from both sides:

−2x=6−6{ -2x = 6 - 6 }

−2x=0{ -2x = 0 }

Divide both sides by -2:

x=0−2{ x = \frac{0}{-2} }

x=0{ x = 0 }

Case 2: -2x+6 = -6

Subtract 6 from both sides:

−2x=−6−6{ -2x = -6 - 6 }

−2x=−12{ -2x = -12 }

Divide both sides by -2:

x=−12−2{ x = \frac{-12}{-2} }

x=6{ x = 6 }

Solving each equation independently is a straightforward algebraic process. In this step, we apply basic algebraic manipulations, such as addition, subtraction, multiplication, and division, to isolate the variable $x$. The key is to perform the same operation on both sides of the equation to maintain balance and ensure the equality remains valid. Each case provides a potential solution for the original absolute value equation. The solutions obtained in this step are critical and must be carefully considered as we move to the final step of verifying the solutions. Accurate execution of these algebraic steps is vital for finding the correct values of $x$. Therefore, double-checking your work in each case is always a good practice when solving absolute value equations.

Step 4: Check the Solutions

It’s always a good practice to check the solutions in the original equation to ensure they are valid. Let's check x = 0 and x = 6.

Check x = 0:

−4∣−2(0)+6∣=−24{ -4|-2(0)+6| = -24 }

−4∣6∣=−24{ -4|6| = -24 }

−4(6)=−24{ -4(6) = -24 }

−24=−24{ -24 = -24 }

This solution is valid.

Check x = 6:

−4∣−2(6)+6∣=−24{ -4|-2(6)+6| = -24 }

−4∣−12+6∣=−24{ -4|-12+6| = -24 }

−4∣−6∣=−24{ -4|-6| = -24 }

−4(6)=−24{ -4(6) = -24 }

−24=−24{ -24 = -24 }

This solution is also valid.

Checking solutions is a crucial step in solving absolute value equations because it verifies that the solutions obtained satisfy the original equation. This step is particularly important in absolute value equations because extraneous solutions can sometimes arise due to the nature of absolute values. By substituting each potential solution back into the original equation, we can confirm whether it makes the equation true. This process helps us avoid errors and ensures the accuracy of our solution. Verification is a best practice in all mathematical problem-solving but is especially critical when dealing with absolute values. Therefore, always take the time to check your solutions to ensure they are correct.

Final Answer

The solutions to the equation $-4|-2x+6|=-24$ are x = 0 and x = 6.

To solidify your understanding, let's recap the key concepts and techniques used in solving the equation.

Isolating the Absolute Value

The first and foremost step is to isolate the absolute value expression. This simplifies the equation and allows us to consider the two possible cases.

Setting Up Two Cases

Due to the nature of absolute value, we always set up two equations:

  1. The expression inside the absolute value equals the positive value on the other side of the equation.
  2. The expression inside the absolute value equals the negative value on the other side of the equation.

Solving Linear Equations

Each case results in a linear equation, which can be solved using standard algebraic techniques.

Checking Solutions

Always check the solutions in the original equation to ensure they are valid and not extraneous.

Common Mistakes to Avoid

When solving absolute value equations, several common mistakes can lead to incorrect solutions. Being aware of these pitfalls can help you avoid them.

Forgetting to Isolate the Absolute Value

Trying to split the equation into cases before isolating the absolute value expression is a common mistake. Always isolate the absolute value first.

Neglecting to Consider Both Cases

Failing to set up and solve both positive and negative cases is a significant error. Remember that the absolute value can result from either a positive or a negative expression.

Incorrectly Solving Linear Equations

Errors in solving the resulting linear equations can lead to wrong answers. Double-check your algebraic manipulations.

Not Checking Solutions

Skipping the step of checking solutions can result in accepting extraneous solutions. Always verify your solutions in the original equation.

Misunderstanding Absolute Value Definition

A fundamental misunderstanding of what absolute value represents (distance from zero) can lead to incorrect problem setup and solutions. Ensure a solid grasp of this concept.

Practice Problems

To further enhance your skills, try solving the following practice problems:

  1. |3x - 2| = 7
  2. 2|x + 1| - 5 = 9
  3. -3|2x - 4| = -18
  4. |4x + 3| = 2x - 1
  5. |x - 5| + 3 = 10

Working through these problems will reinforce the techniques discussed and build your confidence in solving absolute value equations.

Conclusion

Solving absolute value equations involves a systematic approach, including isolating the absolute value, setting up two cases, solving the resulting linear equations, and checking the solutions. By understanding the underlying concepts and avoiding common mistakes, you can confidently solve a wide range of absolute value equations. This article has provided a detailed guide to solving the equation $-4|-2x+6|=-24$, along with the essential knowledge to tackle similar problems. Remember to practice regularly, and you'll master the art of solving absolute value equations in no time. With this understanding, you can tackle more complex mathematical problems with greater confidence. Absolute value is a fundamental concept in mathematics, and mastering its application in equations is a valuable skill. Always remember to approach these problems methodically and carefully, and you will find success.