Solving Absolute Value Equations A Comprehensive Guide To |2x-3|+6=20
In the realm of mathematics, absolute value equations often present a unique challenge that requires a systematic approach to solve. This article delves into the intricacies of solving the equation |2x-3|+6=20, providing a comprehensive guide that not only breaks down the steps but also elucidates the underlying principles. Whether you're a student grappling with algebra or a math enthusiast seeking to expand your knowledge, this detailed explanation will equip you with the tools to tackle similar problems with confidence.
Understanding Absolute Value
Before we dive into the solution, it's crucial to grasp the concept of absolute value. In essence, the absolute value of a number is its distance from zero on the number line. This distance is always non-negative, meaning it's either positive or zero. Mathematically, we denote the absolute value of a number x as |x|. For example, |3| = 3 and |-3| = 3, because both 3 and -3 are three units away from zero. This fundamental understanding is the cornerstone for solving equations involving absolute values.
The Significance of Absolute Value in Equations
The presence of absolute value in an equation introduces a critical fork in the road because it implies two distinct possibilities. Consider the equation |x| = 5. This equation is satisfied by two values: x = 5 and x = -5. Both numbers are five units away from zero. This dual nature is what makes solving absolute value equations a bit more intricate than solving standard linear equations. We must consider both the positive and negative scenarios to ensure we capture all possible solutions.
Deconstructing the Equation |2x-3|+6=20
Now, let's turn our attention to the equation at hand: |2x-3|+6=20. Our primary objective is to isolate the absolute value expression, |2x-3|, on one side of the equation. This is akin to peeling away the layers of an onion to get to the core. The first step in this process involves subtracting 6 from both sides of the equation. This maintains the balance of the equation and moves us closer to isolating the absolute value term. Performing this operation yields:
|2x-3| + 6 - 6 = 20 - 6
Which simplifies to:
|2x-3| = 14
The Two Paths Diverge
With the absolute value expression isolated, we arrive at the pivotal point where we must consider the two possibilities inherent in absolute value equations. The expression inside the absolute value, 2x-3, can be either 14 or -14, as both values have an absolute value of 14. This gives rise to two separate equations:
- 2x - 3 = 14
- 2x - 3 = -14
These two equations represent the two distinct scenarios that satisfy the original absolute value equation. Solving each of these equations will provide us with the solutions to the problem.
Solving the First Equation: 2x-3=14
Let's begin by tackling the first equation: 2x - 3 = 14. This is a standard linear equation that can be solved using basic algebraic techniques. Our goal is to isolate x on one side of the equation. The first step is to eliminate the constant term, -3, by adding 3 to both sides of the equation:
2x - 3 + 3 = 14 + 3
This simplifies to:
2x = 17
Now, to isolate x, we divide both sides of the equation by the coefficient of x, which is 2:
2x / 2 = 17 / 2
This yields the first solution:
x = 17/2
Or, in decimal form:
x = 8.5
Solving the Second Equation: 2x-3=-14
Next, we turn our attention to the second equation: 2x - 3 = -14. This equation mirrors the first but with a negative constant term. We follow the same algebraic steps to isolate x. First, we add 3 to both sides of the equation to eliminate the constant term:
2x - 3 + 3 = -14 + 3
This simplifies to:
2x = -11
Now, we divide both sides of the equation by 2 to isolate x:
2x / 2 = -11 / 2
This gives us the second solution:
x = -11/2
Or, in decimal form:
x = -5.5
Verifying the Solutions
In mathematics, it's always prudent to verify our solutions to ensure accuracy. This is especially important in the case of absolute value equations, where extraneous solutions can sometimes arise. To verify our solutions, we substitute each value of x back into the original equation, |2x-3|+6=20, and check if the equation holds true.
Verification for x = 8.5
Substituting x = 8.5 into the original equation, we get:
|2(8.5) - 3| + 6 = 20
Simplifying the expression inside the absolute value:
|17 - 3| + 6 = 20
|14| + 6 = 20
14 + 6 = 20
20 = 20
The equation holds true, confirming that x = 8.5 is a valid solution.
Verification for x = -5.5
Now, let's substitute x = -5.5 into the original equation:
|2(-5.5) - 3| + 6 = 20
Simplifying the expression inside the absolute value:
|-11 - 3| + 6 = 20
|-14| + 6 = 20
14 + 6 = 20
20 = 20
Again, the equation holds true, verifying that x = -5.5 is also a valid solution.
Conclusion Mastering Absolute Value Equations
In conclusion, solving absolute value equations like |2x-3|+6=20 involves a systematic approach that accounts for the dual nature of absolute value. By isolating the absolute value expression, considering both positive and negative scenarios, and verifying the solutions, we can confidently navigate these mathematical challenges. The solutions to the equation |2x-3|+6=20 are x = 8.5 and x = -5.5. This comprehensive guide has not only demonstrated the steps to solve this specific equation but has also provided a framework for tackling a wide array of absolute value equations. The key takeaway is the importance of understanding the fundamental principles of absolute value and applying them methodically to ensure accurate solutions.
In the realm of algebraic equations, absolute value equations often pose a unique challenge. They require a methodical approach to ensure accurate solutions. This article provides a comprehensive, step-by-step guide on solving the equation |2x-3|+6=20. Whether you are a student learning algebra or someone looking to refresh your math skills, this guide will equip you with the necessary tools and understanding. We will break down the process into manageable steps, making the solution clear and accessible.
Understanding Absolute Value: The Foundation
Before diving into the equation itself, it's crucial to understand the concept of absolute value. The absolute value of a number represents its distance from zero on the number line. This distance is always non-negative. Mathematically, the absolute value of a number x is denoted as |x|. For instance, |5| = 5 and |-5| = 5. This is because both 5 and -5 are five units away from zero. This understanding forms the bedrock for solving absolute value equations.
Why Absolute Value Equations Are Unique
What makes absolute value equations different from standard linear equations is the presence of two potential scenarios. When we encounter |x| = a, where a is a positive number, it means that x can be either a or -a. This duality arises because both a and -a are the same distance from zero. Therefore, solving an absolute value equation involves considering both the positive and negative possibilities of the expression within the absolute value bars.
Isolating the Absolute Value: The First Key Step
Now, let's focus on our specific equation: |2x-3|+6=20. The initial step in solving this equation is to isolate the absolute value expression, which is |2x-3|. This means we need to get the absolute value term by itself on one side of the equation. To do this, we perform the inverse operation of addition, which is subtraction. We subtract 6 from both sides of the equation to maintain balance and isolate the absolute value term:
|2x-3|+6-6=20-6
This simplifies to:
|2x-3|=14
This isolation is a critical milestone. It sets the stage for the next phase, where we consider the two possible cases.
Setting Up the Two Cases: The Fork in the Road
With the absolute value expression isolated, we now face the core concept of absolute value equations: the existence of two possibilities. The expression 2x-3 inside the absolute value bars can be either 14 or -14. This is because the absolute value of both 14 and -14 is 14. This understanding leads us to create two separate equations:
- 2x-3=14
- 2x-3=-14
Each of these equations represents a potential solution path. By solving both equations, we ensure we capture all possible values of x that satisfy the original absolute value equation.
Solving the First Case: 2x-3=14
Let's begin by solving the first equation: 2x-3=14. This is a straightforward linear equation. Our objective is to isolate x on one side of the equation. The first step is to eliminate the constant term, -3, by adding 3 to both sides:
2x-3+3=14+3
This simplifies to:
2x=17
To isolate x, we divide both sides of the equation by the coefficient of x, which is 2:
2x/2=17/2
This yields the first solution:
x=17/2
Or, in decimal form:
x=8.5
Solving the Second Case: 2x-3=-14
Now, let's tackle the second equation: 2x-3=-14. This equation mirrors the first but with a negative constant term. We follow the same algebraic steps to isolate x. First, we add 3 to both sides of the equation:
2x-3+3=-14+3
This simplifies to:
2x=-11
Next, we divide both sides of the equation by 2:
2x/2=-11/2
This gives us the second solution:
x=-11/2
Or, in decimal form:
x=-5.5
Verification: Ensuring Accuracy
In mathematics, it's always a good practice to verify your solutions. This is especially important in absolute value equations, where extraneous solutions can sometimes occur. To verify our solutions, we substitute each value of x back into the original equation, |2x-3|+6=20, and check if the equation holds true.
Verifying x=8.5
Substituting x=8.5 into the original equation, we get:
|2(8.5)-3|+6=20
Simplifying the expression inside the absolute value:
|17-3|+6=20
|14|+6=20
14+6=20
20=20
The equation holds true, confirming that x=8.5 is a valid solution.
Verifying x=-5.5
Now, let's substitute x=-5.5 into the original equation:
|2(-5.5)-3|+6=20
Simplifying the expression inside the absolute value:
|-11-3|+6=20
|-14|+6=20
14+6=20
20=20
Again, the equation holds true, verifying that x=-5.5 is also a valid solution.
Conclusion: Mastering Absolute Value Equations
In conclusion, solving the absolute value equation |2x-3|+6=20 involves a series of logical steps. We first isolate the absolute value expression, then set up two separate equations based on the two possibilities of absolute value, solve each equation, and finally, verify our solutions. The solutions to the equation |2x-3|+6=20 are x=8.5 and x=-5.5. This step-by-step guide provides a clear roadmap for solving similar absolute value equations, enhancing your understanding and confidence in algebraic problem-solving.
Absolute value equations can seem daunting at first, but with a systematic approach, they become manageable. This article offers a detailed explanation on how to solve the absolute value equation |2x-3|+6=20. We will break down each step, providing the reasoning behind it, so that you not only learn the solution but also understand the underlying concepts. Whether you are a student struggling with your homework or someone looking to enhance their mathematical skills, this guide will provide you with a clear and comprehensive understanding of how to solve this type of equation.
The Core Concept Absolute Value Defined
Before we tackle the equation, let's reinforce the core concept of absolute value. The absolute value of a number is its distance from zero on the number line. This distance is always a non-negative value. We denote the absolute value of x as |x|. For example, |7| is 7 because 7 is seven units away from zero, and |-7| is also 7 because -7 is also seven units away from zero. This understanding of distance is the key to unlocking the secrets of absolute value equations.
Why Absolute Value Creates Two Scenarios
The defining characteristic of absolute value equations is that they often lead to two scenarios. This is because the expression inside the absolute value bars can be either positive or negative, and the absolute value will make both results positive. For instance, if we have the equation |x|=4, this means that x could be either 4 or -4, as both numbers have an absolute value of 4. This duality is what makes solving absolute value equations a unique process compared to solving regular linear equations.
Step 1 Isolating the Absolute Value Expression
The first step in solving |2x-3|+6=20 is to isolate the absolute value expression, |2x-3|. This involves getting the absolute value term by itself on one side of the equation. To do this, we need to remove the +6 from the left side. We accomplish this by performing the inverse operation, which is subtraction. We subtract 6 from both sides of the equation to maintain equality:
|2x-3|+6-6=20-6
This simplifies to:
|2x-3|=14
By isolating the absolute value expression, we have set the stage for the next crucial step: considering the two possible cases.
Step 2 Setting Up the Two Possible Cases
Now that we have |2x-3|=14, we need to consider the two possible cases that arise from the absolute value. The expression inside the absolute value bars, 2x-3, can be either equal to 14 or equal to -14. This is because the absolute value of both 14 and -14 is 14. This leads us to set up two separate equations:
- 2x-3=14 (The positive case)
- 2x-3=-14 (The negative case)
By addressing both of these cases, we ensure that we find all possible solutions for x that satisfy the original equation.
Step 3 Solving the First Case 2x-3=14
Let's begin by solving the first case: 2x-3=14. This is a standard linear equation that we can solve using basic algebraic techniques. Our goal is to isolate x. First, we need to eliminate the constant term, -3. We do this by adding 3 to both sides of the equation:
2x-3+3=14+3
This simplifies to:
2x=17
Next, to isolate x, we divide both sides of the equation by the coefficient of x, which is 2:
2x/2=17/2
This gives us our first solution:
x=17/2
Which can also be expressed as a decimal:
x=8.5
Step 4 Solving the Second Case 2x-3=-14
Now, let's move on to solving the second case: 2x-3=-14. This equation is similar to the first case, but it has a negative constant term. We follow the same algebraic steps to isolate x. First, we add 3 to both sides of the equation:
2x-3+3=-14+3
This simplifies to:
2x=-11
Next, we divide both sides of the equation by 2:
2x/2=-11/2
This gives us our second solution:
x=-11/2
Which can also be expressed as a decimal:
x=-5.5
Step 5 Verifying the Solutions Accuracy is Key
In mathematics, it's always crucial to verify your solutions, especially in absolute value equations, where extraneous solutions can sometimes arise. To verify our solutions, we substitute each value of x back into the original equation, |2x-3|+6=20, and check if the equation holds true.
Verification for x=8.5
Substituting x=8.5 into the original equation, we get:
|2(8.5)-3|+6=20
Simplifying the expression inside the absolute value:
|17-3|+6=20
|14|+6=20
14+6=20
20=20
The equation holds true, confirming that x=8.5 is a valid solution.
Verification for x=-5.5
Now, let's substitute x=-5.5 into the original equation:
|2(-5.5)-3|+6=20
Simplifying the expression inside the absolute value:
|-11-3|+6=20
|-14|+6=20
14+6=20
20=20
Again, the equation holds true, verifying that x=-5.5 is also a valid solution.
Conclusion Mastering the Art of Solving Absolute Value Equations
In conclusion, solving the absolute value equation |2x-3|+6=20 involves a step-by-step process that begins with isolating the absolute value expression, then branching into two possible cases, solving each case separately, and finally, verifying the solutions. The solutions to the equation |2x-3|+6=20 are x=8.5 and x=-5.5. By understanding the core concept of absolute value and following this detailed explanation, you can confidently tackle similar absolute value equations and master this important algebraic skill. This guide not only provides the solution but also aims to enhance your understanding of the underlying principles, empowering you to approach mathematical problems with clarity and confidence.
