Solve The Equation: $\[ 2|x| - 3 = 8 \\]

Introduction

Absolute value equations are a fundamental concept in mathematics, and solving them requires a clear understanding of the properties of absolute value. In this article, we will focus on solving the equation $2|x| - 3 = 8$, which involves isolating the absolute value expression and then considering the two possible cases that arise from it.

Understanding Absolute Value

Before we dive into solving the equation, let's take a moment to understand what absolute value is. The absolute value of a number $x$, denoted by $|x|$, is the distance of $x$ from zero on the number line. In other words, it is the magnitude of $x$ without considering its direction. For example, the absolute value of $-3$ is $3$, and the absolute value of $4$ is also $4$.

The Equation $2|x| - 3 = 8$

Now that we have a good understanding of absolute value, let's tackle the equation $2|x| - 3 = 8$. Our goal is to isolate the absolute value expression and then solve for $x$. To do this, we will follow a series of steps that will help us simplify the equation and ultimately find the solution.

Step 1: Add 3 to Both Sides

The first step in solving the equation is to add 3 to both sides, which will help us isolate the absolute value expression. This gives us:

$$2|x| = 8 + 3$$

$$2|x| = 11$$

Step 2: Divide Both Sides by 2

Next, we will divide both sides of the equation by 2, which will help us further isolate the absolute value expression. This gives us:

$$|x| = \frac{11}{2}$$

Step 3: Consider the Two Possible Cases

Now that we have isolated the absolute value expression, we need to consider the two possible cases that arise from it. These cases are:

• $x \geq 0$
• $x < 0$

Case 1: $x \geq 0$

If $x \geq 0$, then the absolute value of $x$ is simply $x$. Therefore, we can write:

$$x = \frac{11}{2}$$

This is the solution to the equation when $x \geq 0$.

Case 2: $x < 0$

If $x < 0$, then the absolute value of $x$ is $-x$. Therefore, we can write:

$$-x = \frac{11}{2}$$

Solving for $x$, we get:

$$x = -\frac{11}{2}$$

This is the solution to the equation when $x < 0$.

Conclusion

In this article, we have solved the equation $2|x| - 3 = 8$ by isolating the absolute value expression and then considering the two possible cases that arise from it. We have shown that the solution to the equation is $x = \frac{11}{2}$ when $x \geq 0$ and $x = -\frac{11}{2}$ when $x < 0$. By following these steps, we have demonstrated how to solve absolute value equations and have gained a deeper understanding of the properties of absolute value.

Absolute Value Equations: Key Concepts

• Definition of Absolute Value: The absolute value of a number $x$, denoted by $|x|$, is the distance of $x$ from zero on the number line.
• Properties of Absolute Value: The absolute value of a number is always non-negative, and the absolute value of a product or quotient is the product or quotient of the absolute values.
• Solving Absolute Value Equations: To solve an absolute value equation, we need to isolate the absolute value expression and then consider the two possible cases that arise from it.

Real-World Applications of Absolute Value Equations

Absolute value equations have numerous real-world applications, including:

• Physics: In physics, absolute value equations are used to model the motion of objects and to describe the behavior of physical systems.
• Engineering: In engineering, absolute value equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: In economics, absolute value equations are used to model economic systems and to analyze the behavior of economic variables.

Conclusion

Q: What is an absolute value equation?

A: An absolute value equation is an equation that involves the absolute value of a variable or expression. The absolute value of a number is its distance from zero on the number line, without considering its direction.

Q: How do I solve an absolute value equation?

A: To solve an absolute value equation, you need to isolate the absolute value expression and then consider the two possible cases that arise from it. This involves adding or subtracting the same value to both sides of the equation, and then solving for the variable.

Q: What are the two possible cases that arise from an absolute value equation?

A: The two possible cases that arise from an absolute value equation are:

• $x \geq 0$: In this case, the absolute value of $x$ is simply $x$.
• $x < 0$: In this case, the absolute value of $x$ is $-x$.

Q: How do I determine which case to use?

A: To determine which case to use, you need to consider the sign of the variable. If the variable is non-negative, you use the first case. If the variable is negative, you use the second case.

Q: What are some common mistakes to avoid when solving absolute value equations?

A: Some common mistakes to avoid when solving absolute value equations include:

• Failing to isolate the absolute value expression
• Not considering both possible cases
• Not checking the sign of the variable

Q: How do I check my answer to an absolute value equation?

A: To check your answer to an absolute value equation, you need to plug your solution back into the original equation and verify that it is true. This involves substituting your solution for the variable and simplifying the equation.

Q: What are some real-world applications of absolute value equations?

A: Absolute value equations have numerous real-world applications, including:

• Physics: In physics, absolute value equations are used to model the motion of objects and to describe the behavior of physical systems.
• Engineering: In engineering, absolute value equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: In economics, absolute value equations are used to model economic systems and to analyze the behavior of economic variables.

Q: How do I graph an absolute value equation?

A: To graph an absolute value equation, you need to plot the two possible cases that arise from it. This involves plotting the graph of the first case and then reflecting it across the y-axis to obtain the graph of the second case.

Q: What are some common types of absolute value equations?

A: Some common types of absolute value equations include:

• Linear absolute value equations: These are absolute value equations that involve a linear expression.
• Quadratic absolute value equations: These are absolute value equations that involve a quadratic expression.
• Polynomial absolute value equations: These are absolute value equations that involve a polynomial expression.

Q: How do I solve a system of absolute value equations?

A: To solve a system of absolute value equations, you need to isolate the absolute value expressions and then consider the two possible cases that arise from each equation. This involves using substitution or elimination to solve the system.

Conclusion

In conclusion, absolute value equations are a fundamental concept in mathematics, and solving them requires a clear understanding of the properties of absolute value. By following the steps outlined in this article, we have demonstrated how to solve absolute value equations and have gained a deeper understanding of the properties of absolute value. We have also highlighted the real-world applications of absolute value equations and have shown how they are used in physics, engineering, and economics.