Analyze The Solution Shown:1. { -|-x|=7$}$ : Given2. { |-x|=-7$}$ : Multiplication Property Of Equality3. { -x=7$}$ Or { -x=-7$}$ : Definition Of Absolute Value4. { X=-7$}$ Or { X=7$}$ :

Mar 13, 2025 by ADMIN 187 views

**Solving Absolute Value Equations: A Step-by-Step Analysis**

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 analyze a given solution to an absolute value equation and break down the steps involved in solving it. We will also discuss the properties of absolute value and how they are applied in solving absolute value equations.

The Given Equation

The given equation is:

|-x|=7

This equation involves the absolute value of a variable $x$ , and our goal is to solve for $x$ .

Step 1: Multiplication Property of Equality

The first step in solving the equation is to apply the multiplication property of equality. This property states that if two expressions are equal, then their products with the same non-zero number are also equal. In this case, we can multiply both sides of the equation by $-1$ to get:

|-x|=-7

This step is valid because we are multiplying both sides of the equation by the same non-zero number, which is $-1$ .

Step 2: Definition of Absolute Value

The next step is to apply the definition of absolute value. The absolute value of a number $x$ is defined as:

|x|=\begin{cases}x, & \text{if }x\geq0\\-x, & \text{if }x<0\end{cases}

In this case, we have $|-x|=-7$ , which means that either $-x=7$ or $-x=-7$ .

Step 3: Solving for x

Now that we have two possible equations, we can solve for $x$ by isolating it on one side of the equation. In the first case, we have:

-x=7

Multiplying both sides by $-1$ , we get:

x=-7

In the second case, we have:

-x=-7

Multiplying both sides by $-1$ , we get:

x=7

Conclusion

In conclusion, we have solved the absolute value equation $|-x|=7$ by applying the multiplication property of equality and the definition of absolute value. We have found two possible solutions for $x$ , which are $x=-7$ and $x=7$ . These solutions are valid because they satisfy the original equation.

Properties of Absolute Value

Absolute value has several important properties that are used in solving absolute value equations. Some of these properties include:

Non-Negativity: The absolute value of a number is always non-negative, i.e., $|x|\geq0$ for all $x$ .
Identity of Indiscernibles: If $|x|=|y|$ , then $x=y$ or $x=-y$ .
Triangle Inequality: $|x+y|\leq|x|+|y|$ for all $x$ and $y$ .
Multiplication Property of Equality: If $|x|=|y|$ , then $x=y$ or $x=-y$ .

These properties are essential in solving absolute value equations and are used extensively in mathematics and other fields.

Applications of Absolute Value

Absolute value has numerous applications in mathematics and other fields. Some of these applications include:

Distance and Length: Absolute value is used to measure distance and length in mathematics and physics.
Finance: Absolute value is used in finance to calculate the absolute return on investment.
Computer Science: Absolute value is used in computer science to calculate the absolute difference between two numbers.
Engineering: Absolute value is used in engineering to calculate the absolute error in a measurement.

In conclusion, absolute value is a fundamental concept in mathematics that has numerous applications in various fields. Solving absolute value equations requires a clear understanding of the properties of absolute value and how they are applied in solving these equations.

Final Thoughts

Introduction

In our previous article, we analyzed a given solution to an absolute value equation and broke down the steps involved in solving it. We also discussed the properties of absolute value and how they are applied in solving absolute value equations. In this article, we will provide a Q&A guide to help you better understand absolute value equations and how to solve them.

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 direction.

Q: How do I solve an absolute value equation?

A: To solve an absolute value equation, you need to follow these steps:

Isolate the absolute value expression on one side of the equation.
Set up two equations: one where the expression inside the absolute value is positive, and one where it is negative.
Solve each equation separately.
Check your solutions to make sure they satisfy the original equation.

Q: What are the properties of absolute value?

A: The properties of absolute value are:

Non-Negativity: The absolute value of a number is always non-negative, i.e., $|x|\geq0$ for all $x$ .
Identity of Indiscernibles: If $|x|=|y|$ , then $x=y$ or $x=-y$ .
Triangle Inequality: $|x+y|\leq|x|+|y|$ for all $x$ and $y$ .
Multiplication Property of Equality: If $|x|=|y|$ , then $x=y$ or $x=-y$ .

Q: How do I apply the multiplication property of equality in solving absolute value equations?

A: To apply the multiplication property of equality, you need to multiply both sides of the equation by the same non-zero number. This will help you isolate the absolute value expression and set up two equations.

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:

Not isolating the absolute value expression: Make sure to isolate the absolute value expression on one side of the equation.
Not setting up two equations: Make sure to set up two equations: one where the expression inside the absolute value is positive, and one where it is negative.
Not checking solutions: Make sure to check your solutions to make sure they satisfy the original equation.

Q: How do I check my solutions to make sure they satisfy the original equation?

A: To check your solutions, you need to plug them back into the original equation and make sure they satisfy it. If they do, then you have found a valid solution.

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

A: Some real-world applications of absolute value equations include:

Distance and length: Absolute value is used to measure distance and length in mathematics and physics.
Finance: Absolute value is used in finance to calculate the absolute return on investment.
Computer science: Absolute value is used in computer science to calculate the absolute difference between two numbers.
Engineering: Absolute value is used in engineering to calculate the absolute error in a measurement.

Conclusion

In conclusion, absolute value equations are an important concept in mathematics that have numerous applications in various fields. By following the steps outlined in this article and avoiding common mistakes, you can solve absolute value equations with confidence. Remember to isolate the absolute value expression, set up two equations, and check your solutions to make sure they satisfy the original equation.