Which Of The Following Is(are) The Solution(s) To ∣ 3 X + 10 ∣ = 10 |3x + 10| = 10 ∣3 X + 10∣ = 10 ?A. X = 0 X = 0 X = 0 B. X = − 20 3 , 0 X = -\frac{20}{3}, 0 X = − 3 20 ​ , 0 C. X = 20 3 X = \frac{20}{3} X = 3 20 ​ D. X = 20 3 , 0 X = \frac{20}{3}, 0 X = 3 20 ​ , 0

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 $|3x + 10| = 10$, which is a classic example of an absolute value equation. We will explore the different solutions to this equation and provide a step-by-step guide on how to arrive at the correct answer.

Understanding Absolute Value

Before we dive into solving the equation, let's take a moment to understand what absolute value means. The absolute value of a number is its distance from zero on the number line. In other words, it is the magnitude of the number without considering its sign. For example, the absolute value of 5 is 5, and the absolute value of -5 is also 5.

The Equation $|3x + 10| = 10$

Now that we have a good understanding of absolute value, let's focus on the equation $|3x + 10| = 10$. This equation states that the absolute value of $3x + 10$ is equal to 10. To solve this equation, we need to consider two cases: when $3x + 10$ is positive and when $3x + 10$ is negative.

Case 1: $3x + 10 \geq 0$

When $3x + 10 \geq 0$, we can remove the absolute value sign and solve the equation as follows:

$3x + 10 = 10$

Subtracting 10 from both sides gives us:

$3x = 0$

Dividing both sides by 3 gives us:

$x = 0$

So, one solution to the equation is $x = 0$.

Case 2: $3x + 10 < 0$

When $3x + 10 < 0$, we need to remove the absolute value sign and multiply the expression by -1 to make it positive. This gives us:

$-(3x + 10) = 10$

Expanding the left-hand side gives us:

$-3x - 10 = 10$

Adding 10 to both sides gives us:

$-3x = 20$

Dividing both sides by -3 gives us:

$x = -\frac{20}{3}$

So, another solution to the equation is $x = -\frac{20}{3}$.

Combining the Solutions

We have found two solutions to the equation: $x = 0$ and $x = -\frac{20}{3}$. However, we need to check if these solutions satisfy the original equation.

Substituting $x = 0$ into the original equation gives us:

$|3(0) + 10| = 10$

Simplifying the left-hand side gives us:

$|10| = 10$

This is true, so $x = 0$ is a valid solution.

Substituting $x = -\frac{20}{3}$ into the original equation gives us:

$|3(-\frac{20}{3}) + 10| = 10$

Simplifying the left-hand side gives us:

$|-20 + 10| = 10$

Simplifying further gives us:

$|-10| = 10$

This is also true, so $x = -\frac{20}{3}$ is a valid solution.

Conclusion

In conclusion, the solutions to the equation $|3x + 10| = 10$ are $x = 0$ and $x = -\frac{20}{3}$. These solutions satisfy the original equation and are valid.

Final Answer

The final answer is:

B. $x = -\frac{20}{3}, 0$

Additional Tips and Tricks

• When solving absolute value equations, always consider two cases: when the expression inside the absolute value is positive and when it is negative.
• When removing the absolute value sign, make sure to consider the sign of the expression inside the absolute value.
• When multiplying an expression by -1, make sure to change the sign of the expression.

Introduction

In our previous article, we explored the concept of absolute value equations and solved the equation $|3x + 10| = 10$. 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 a number is its distance from zero on the number line. In other words, it is the magnitude of the number without considering its sign.

Q: How do I solve an absolute value equation?

A: To solve an absolute value equation, you need to consider two cases: when the expression inside the absolute value is positive and when it is negative. You can remove the absolute value sign and solve the equation as follows:

• When the expression inside the absolute value is positive, you can remove the absolute value sign and solve the equation as usual.
• When the expression inside the absolute value is negative, you need to remove the absolute value sign and multiply the expression by -1 to make it positive.

Q: What are the two cases I need to consider when solving an absolute value equation?

A: The two cases you need to consider when solving an absolute value equation are:

• Case 1: When the expression inside the absolute value is greater than or equal to zero.
• Case 2: When the expression inside the absolute value is less than zero.

Q: How do I determine which case to use?

A: To determine which case to use, you need to check the sign of the expression inside the absolute value. If the expression is positive, you can use Case 1. If the expression is negative, you can use Case 2.

Q: What is the difference between Case 1 and Case 2?

A: The main difference between Case 1 and Case 2 is the sign of the expression inside the absolute value. In Case 1, the expression is positive, and you can remove the absolute value sign and solve the equation as usual. In Case 2, the expression is negative, and you need to remove the absolute value sign and multiply the expression by -1 to make it positive.

Q: Can I use a calculator to solve an absolute value equation?

A: Yes, you can use a calculator to solve an absolute value equation. However, it's always a good idea to check your work by hand to make sure you understand the solution.

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 considering both cases (positive and negative)
• Not removing the absolute value sign correctly
• Not multiplying the expression by -1 when necessary
• Not checking the sign of the expression inside the absolute value

Q: How can I practice solving absolute value equations?

A: You can practice solving absolute value equations by working through examples and exercises. You can also use online resources, such as Khan Academy or Mathway, to help you practice.

Conclusion

In conclusion, absolute value equations can be challenging to solve, but with practice and patience, you can master them. Remember to consider both cases (positive and negative), remove the absolute value sign correctly, and multiply the expression by -1 when necessary. By following these tips and practicing regularly, you can become proficient in solving absolute value equations.

Additional Resources

• Khan Academy: Absolute Value Equations
• Mathway: Absolute Value Equations
• IXL: Absolute Value Equations

By using these resources and practicing regularly, you can improve your skills and become more confident in solving absolute value equations.