Solve $|x+5|=-8$.A. $x=-3$ And $x=13$ B. No Solution C. $x=3$ And $x=-13$ D. $x=-3$ And $x=-13$

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Introduction

When dealing with absolute value equations, we need to consider both the positive and negative cases. In this case, we have the equation $|x+5|=-8$. Our goal is to find the value of $x$ that satisfies this equation. We will break down the solution step by step and explore the different possibilities.

Understanding Absolute Value Equations

Absolute value equations are equations that involve 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. In other words, the absolute value of a number is always non-negative.

When solving absolute value equations, we need to consider two cases:

1. The expression inside the absolute value is positive.
2. The expression inside the absolute value is negative.

Case 1: The Expression Inside the Absolute Value is Positive

In this case, we have $x+5 \geq 0$. This means that $x \geq -5$. Now, we can rewrite the equation as $x+5=-8$. To solve for $x$, we subtract 5 from both sides of the equation:

$$x+5-5=-8-5$$

$$x=-13$$

Case 2: The Expression Inside the Absolute Value is Negative

In this case, we have $x+5 < 0$. This means that $x < -5$. Now, we can rewrite the equation as $-(x+5)=-8$. To solve for $x$, we distribute the negative sign and then simplify:

$$-(x+5)=-8$$

$$-x-5=-8$$

$$-x=-8+5$$

$$-x=-3$$

$$x=3$$

Combining the Results

We have found two possible values for $x$: $x=-13$ and $x=3$. However, we need to check if these values satisfy the original equation.

Checking the Solutions

Let's substitute $x=-13$ into the original equation:

$$|(-13)+5|=-8$$

$$|-8|=-8$$

This is not true, since the absolute value of $-8$ is $8$, not $-8$. Therefore, $x=-13$ is not a solution.

Now, let's substitute $x=3$ into the original equation:

$$|(3)+5|=-8$$

$$|8|=-8$$

This is not true, since the absolute value of $8$ is $8$, not $-8$. Therefore, $x=3$ is not a solution.

Conclusion

We have found that neither $x=-13$ nor $x=3$ is a solution to the equation $|x+5|=-8$. This means that the equation has no solution.

The final answer is: B. No solution

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Introduction

In our previous article, we solved the equation $|x+5|=-8$ and found that it has no solution. However, we received many questions from readers who were unsure about the steps involved in solving absolute value equations. In this article, we will answer some of the most frequently asked questions about solving absolute value equations.

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 consider two cases:

1. The expression inside the absolute value is positive.
2. The expression inside the absolute value is negative.

Q: What if the expression inside the absolute value is positive?

A: If the expression inside the absolute value is positive, you can simply remove the absolute value signs and solve the resulting equation.

Q: What if the expression inside the absolute value is negative?

A: If the expression inside the absolute value is negative, you need to multiply the expression by $-1$ and then remove the absolute value signs. This is because the absolute value of a negative number is its positive counterpart.

Q: How do I know 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, use case 1. If the expression is negative, use case 2.

Q: What if the equation has no solution?

A: If the equation has no solution, it means that there is no value of the variable that can satisfy the equation. This can happen if the equation is inconsistent, meaning that it is impossible to find a value of the variable that makes the equation true.

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

A: Yes, you can use a calculator to solve absolute value equations. However, it's always a good idea to check your work by hand to make sure that the calculator is giving you the correct answer.

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 checking the sign of the expression inside the absolute value
• Not removing the absolute value signs correctly
• Not checking the solution to make sure it satisfies the original equation

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

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

Q: What are some examples of absolute value equations in real-world problems?

A: Some examples of absolute value equations in real-world problems include:

• A company has a budget of $1000 to spend on advertising. If they spend more than $1000, they will be penalized. Write an absolute value equation to model this situation.
• A person is driving a car and needs to arrive at a destination within a certain time limit. If they arrive late, they will be fined. Write an absolute value equation to model this situation.
• A thermometer is measuring the temperature of a room. If the temperature is above a certain threshold, the air conditioning will turn on. Write an absolute value equation to model this situation.

Conclusion

Solving absolute value equations can be a challenging task, but with practice and patience, you can become proficient in solving them. Remember to consider both cases (positive and negative), check the sign of the expression inside the absolute value, and remove the absolute value signs correctly. With these tips and examples, you'll be well on your way to becoming an expert in solving absolute value equations.