Select The Correct Answer.How Many Solutions Exist For 2 ∣ X − 1 ∣ = 8 2|x-1|=8 2∣ X − 1∣ = 8 ?A. 0 B. 1 C. 2 D. 3

Introduction

Absolute value equations are a fundamental concept in algebra, and solving them requires a clear understanding of the properties of absolute value. In this article, we will focus on solving the equation $2|x-1|=8$ and determine the number of solutions that exist. We will break down the solution process into manageable steps, making it easier to understand and apply.

Understanding Absolute Value

Before we dive into solving the equation, let's review the concept of absolute value. 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, $|5| = 5$ and $|-5| = 5$.

Solving the Equation

Now that we have a good understanding of absolute value, let's solve the equation $2|x-1|=8$. To do this, we will follow these steps:

Step 1: Isolate the Absolute Value Expression

The first step is to isolate the absolute value expression on one side of the equation. We can do this by dividing both sides of the equation by 2:

$$|x-1| = 4$$

Step 2: Write Two Separate Equations

Since the absolute value of an expression can be positive or negative, we need to write two separate equations:

$$x-1 = 4 \quad \text{or} \quad x-1 = -4$$

Step 3: Solve Each Equation

Now that we have two separate equations, we can solve each one:

Equation 1: $x-1 = 4$

$$x = 4 + 1$$

$$x = 5$$

Equation 2: $x-1 = -4$

$$x = -4 + 1$$

$$x = -3$$

Conclusion

We have found two solutions to the equation $2|x-1|=8$: $x = 5$ and $x = -3$. Therefore, the correct answer is:

C. 2

Why Two Solutions?

You may be wondering why we have two solutions to the equation. The reason is that the absolute value of an expression can be positive or negative. In this case, the absolute value of $x-1$ can be either 4 or -4, resulting in two separate equations.

Tips and Tricks

When solving absolute value equations, remember to:

• Isolate the absolute value expression on one side of the equation
• Write two separate equations
• Solve each equation separately
• Check your solutions to make sure they satisfy the original equation

By following these steps and tips, you will be able to solve absolute value equations with ease.

Common Mistakes

When solving absolute value equations, some common mistakes to avoid include:

• Not isolating the absolute value expression
• Not writing two separate equations
• Not checking solutions to make sure they satisfy the original equation

By being aware of these common mistakes, you can avoid them and ensure that your solutions are correct.

Conclusion

$$$$$$

Introduction

In our previous article, we solved the equation $2|x-1|=8$ and determined that there are two solutions: $x = 5$ and $x = -3$. 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 contains an absolute value expression. The absolute value of an expression 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, follow these steps:

1. Isolate the absolute value expression on one side of the equation.
2. Write two separate equations: one where the absolute value expression is positive and one where it is negative.
3. Solve each equation separately.
4. Check your solutions to make sure they satisfy the original equation.

Q: Why do I need to write two separate equations?

A: You need to write two separate equations because the absolute value of an expression can be positive or negative. By writing two separate equations, you can account for both possibilities.

Q: What if I have a negative number inside the absolute value expression?

A: If you have a negative number inside the absolute value expression, you can rewrite it as a positive number by multiplying it by -1. For example, $|x-1| = 4$ is equivalent to $|-(x-1)| = 4$.

Q: Can I have more than two solutions to an absolute value equation?

A: No, you cannot have more than two solutions to an absolute value equation. This is because the absolute value of an expression can only be positive or negative, resulting in at most two separate equations.

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

A: To check your solutions, plug them back into the original equation and make sure they are true. For example, if you have the solution $x = 5$, plug it back into the equation $2|x-1|=8$ to make sure it is true.

Q: What if I have a fraction or decimal inside the absolute value expression?

A: If you have a fraction or decimal inside the absolute value expression, you can simplify it by multiplying both sides of the equation by the denominator or by converting it to a decimal.

Q: Can I use absolute value equations to solve real-world problems?

A: Yes, you can use absolute value equations to solve real-world problems. For example, you can use them to model the distance between two objects, the amount of money you have in your bank account, or the temperature in a room.

Conclusion

In this Q&A guide, we have provided answers to common questions about absolute value equations and how to solve them. By following the steps outlined in this guide, you will be able to solve absolute value equations with confidence and apply them to real-world problems.

Common Mistakes to Avoid

When solving absolute value equations, some common mistakes to avoid include:

• Not isolating the absolute value expression
• Not writing two separate equations
• Not checking solutions to make sure they satisfy the original equation
• Not simplifying fractions or decimals inside the absolute value expression

By being aware of these common mistakes, you can avoid them and ensure that your solutions are correct.

Tips and Tricks

When solving absolute value equations, remember to:

• Isolate the absolute value expression on one side of the equation
• Write two separate equations
• Solve each equation separately
• Check your solutions to make sure they satisfy the original equation
• Simplify fractions or decimals inside the absolute value expression

By following these tips and tricks, you will be able to solve absolute value equations with ease and apply them to real-world problems.