Solving Absolute Value Equations A Step-by-Step Guide To |3x-4(x+1)|=2

In the realm of mathematics, solving equations involving absolute values can often seem daunting. However, with a systematic approach and a clear understanding of the properties of absolute values, these equations can be tackled with confidence. In this comprehensive guide, we will delve into the step-by-step process of solving the equation |3x-4(x+1)|=2, illustrating the solution set on a number line and providing valuable insights into the underlying concepts.

Understanding Absolute Value Equations

Before we embark on solving the equation at hand, it's crucial to grasp the fundamental concept of absolute value. The absolute value of a number represents its distance from zero on the number line, irrespective of direction. Consequently, the absolute value of a number 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.

When dealing with equations involving absolute values, we encounter a unique situation where the expression inside the absolute value can be either positive or negative, yet its absolute value remains the same. This principle forms the cornerstone of solving absolute value equations. To illustrate, consider the equation |x| = 3. This equation has two solutions: x = 3 and x = -3, because both 3 and -3 are at a distance of 3 units from zero on the number line. This fundamental understanding sets the stage for solving more complex absolute value equations.

Deconstructing the Equation: |3x-4(x+1)|=2

Now, let's turn our attention to the specific equation at hand: |3x-4(x+1)|=2. This equation involves an absolute value expression on one side and a constant on the other. To solve it effectively, we need to consider two distinct scenarios: one where the expression inside the absolute value is positive or zero, and another where it is negative. This bifurcation stems from the very definition of absolute value, which dictates that |a| equals a if a is non-negative and -a if a is negative.

Our first step involves simplifying the expression inside the absolute value. We begin by distributing the -4 across the parentheses: 3x - 4(x + 1) = 3x - 4x - 4. Combining like terms, we arrive at the simplified expression -x - 4. Thus, our original equation |3x-4(x+1)|=2 transforms into |-x-4|=2. This simplification lays the groundwork for the subsequent steps in solving the equation.

Case 1: The Expression Inside the Absolute Value is Non-Negative

Let's consider the first scenario: where the expression inside the absolute value, -x - 4, is greater than or equal to zero. In mathematical notation, this is represented as -x - 4 ≥ 0. When this condition holds true, the absolute value of the expression is simply the expression itself. Consequently, the equation |-x-4|=2 becomes -x - 4 = 2. Now, we have a linear equation that we can solve for x.

To isolate x, we first add 4 to both sides of the equation, resulting in -x = 6. Next, we multiply both sides by -1 to obtain x = -6. This value of x represents a potential solution to the original absolute value equation, but it's crucial to verify whether it satisfies the initial condition -x - 4 ≥ 0. Substituting x = -6 into the inequality, we get -(-6) - 4 ≥ 0, which simplifies to 6 - 4 ≥ 0, or 2 ≥ 0. This inequality holds true, confirming that x = -6 is indeed a valid solution.

Case 2: The Expression Inside the Absolute Value is Negative

Now, let's explore the second scenario: where the expression inside the absolute value, -x - 4, is less than zero. Mathematically, this is expressed as -x - 4 < 0. When this condition is met, the absolute value of the expression is the negation of the expression. Therefore, the equation |-x-4|=2 transforms into -(-x - 4) = 2. Simplifying the left side of the equation, we get x + 4 = 2.

To solve for x, we subtract 4 from both sides of the equation, yielding x = -2. This value of x represents another potential solution to the original absolute value equation. Again, it's essential to verify whether this solution satisfies the initial condition -x - 4 < 0. Substituting x = -2 into the inequality, we get -(-2) - 4 < 0, which simplifies to 2 - 4 < 0, or -2 < 0. This inequality holds true, confirming that x = -2 is also a valid solution.

Solution Set and Number Line Representation

Having solved the equation |3x-4(x+1)|=2, we have identified two distinct solutions: x = -6 and x = -2. The solution set, which represents the collection of all values of x that satisfy the equation, is denoted as {-6, -2}. To provide a visual representation of the solution set, we can plot these values on a number line.

On the number line, we draw a horizontal line and mark the points corresponding to -6 and -2. Since these points are solutions to the equation, we represent them with closed circles or dots. The remaining portions of the number line, which do not correspond to solutions, are left unmarked. This visual representation offers a clear and concise depiction of the solution set, making it easier to grasp the concept of solutions to absolute value equations.

Visualizing the Solution Set on a Number Line

To visualize the solution set {-6, -2} on a number line, follow these steps:

1. Draw a horizontal line: This line represents the number line, extending infinitely in both directions.
2. Mark zero: Choose a point on the line to represent zero. This serves as a reference point for locating other numbers.
3. Mark the solutions: Locate -6 and -2 on the number line. Since -6 is less than -2, it will be to the left of -2. Mark these points with closed circles or dots to indicate that they are included in the solution set.
4. Indicate the solution set: The closed circles or dots at -6 and -2 visually represent the solution set. There are no other points on the number line that satisfy the equation, so the rest of the line remains unmarked.

This number line representation provides a clear visual understanding of the solution set, showing exactly which values of x satisfy the original equation.

Key Takeaways for Solving Absolute Value Equations

Solving equations involving absolute values requires a systematic approach and careful consideration of the properties of absolute values. Here are some key takeaways to keep in mind:

• Isolate the absolute value expression: Before proceeding, isolate the absolute value expression on one side of the equation.
• Consider two cases: Recognize that the expression inside the absolute value can be either positive or negative. Set up two separate equations to address each case.
• Solve each case: Solve each equation independently, using standard algebraic techniques.
• Verify solutions: It's crucial to verify whether the solutions obtained in each case satisfy the original equation. This step ensures that you have identified all valid solutions.
• Represent the solution set: Express the solution set as a collection of values that satisfy the equation. You can also represent the solution set visually on a number line.

By adhering to these guidelines, you can confidently tackle a wide range of absolute value equations.

Conclusion: Mastering Absolute Value Equations

Solving equations involving absolute values may initially appear challenging, but with a clear understanding of the underlying principles and a systematic approach, these equations can be solved with proficiency. In this guide, we have meticulously dissected the process of solving the equation |3x-4(x+1)|=2, emphasizing the importance of considering both positive and negative scenarios for the expression inside the absolute value. Furthermore, we have illustrated the solution set on a number line, providing a visual representation of the values that satisfy the equation.

By mastering the techniques outlined in this guide, you will be well-equipped to confidently solve a variety of absolute value equations, further solidifying your understanding of fundamental mathematical concepts.