What Is The Solution Of $|x+7| \leq 15$?A. $-8 \leq X \leq 22$B. $-22 \leq X \leq 8$C. $x \leq -8$ Or $x \geq 22$D. $x \leq -22$ Or $x \geq 8$

Introduction

In mathematics, absolute value equations are a type of inequality that involves the absolute value of a variable or expression. These equations are used to represent the distance of a number from zero on the number line. In this article, we will explore the solution of the absolute value equation $|x+7| \leq 15$. We will break down the solution step by step and provide a clear explanation of the process.

Understanding Absolute Value Equations

Absolute value equations are of the form $|x-a| \leq b$, where $a$ and $b$ are constants. The absolute value of a number is its distance from zero on the number line. For example, the absolute value of $-3$ is $3$, because $-3$ is $3$ units away from zero on the number line.

Solving Absolute Value Equations

To solve an absolute value equation, we need to consider two cases:

• Case 1: $x-a \geq 0$
• Case 2: $x-a < 0$

For case 1, the absolute value equation becomes $x-a \leq b$. For case 2, the absolute value equation becomes $-(x-a) \leq b$.

Solving the Equation $|x+7| \leq 15$

We will use the two-case method to solve the equation $|x+7| \leq 15$. First, we will consider case 1, where $x+7 \geq 0$. In this case, the equation becomes $x+7 \leq 15$. Subtracting 7 from both sides, we get $x \leq 8$.

Next, we will consider case 2, where $x+7 < 0$. In this case, the equation becomes $-(x+7) \leq 15$. Simplifying, we get $-x-7 \leq 15$. Adding 7 to both sides, we get $-x \leq 22$. Multiplying both sides by -1, we get $x \geq -22$.

Combining the Solutions

We have found two solutions: $x \leq 8$ and $x \geq -22$. However, we need to combine these solutions to find the final answer. Since the two solutions overlap, we can combine them to get the final solution: $-22 \leq x \leq 8$.

Conclusion

In this article, we have solved the absolute value equation $|x+7| \leq 15$ using the two-case method. We have found the solution to be $-22 \leq x \leq 8$. This solution represents the range of values of $x$ that satisfy the equation.

Final Answer

The final answer is $-22 \leq x \leq 8$. This is the correct solution to the equation $|x+7| \leq 15$.

Comparison of Options

Let's compare our solution with the given options:

• Option A: $-8 \leq x \leq 22$
• Option B: $-22 \leq x \leq 8$
• Option C: $x \leq -8$ or $x \geq 22$
• Option D: $x \leq -22$ or $x \geq 8$

Our solution matches option B: $-22 \leq x \leq 8$. Therefore, the correct answer is option B.

Frequently Asked Questions

• What is the solution to the equation $|x+7| \leq 15$?
• How do I solve an absolute value equation?
• What is the two-case method for solving absolute value equations?

Step-by-Step Solution

Here is the step-by-step solution to the equation $|x+7| \leq 15$:

1. Consider case 1: $x+7 \geq 0$
2. Solve the equation $x+7 \leq 15$
3. Subtract 7 from both sides to get $x \leq 8$
4. Consider case 2: $x+7 < 0$
5. Solve the equation $-(x+7) \leq 15$
6. Simplify to get $-x-7 \leq 15$
7. Add 7 to both sides to get $-x \leq 22$
8. Multiply both sides by -1 to get $x \geq -22$
9. Combine the solutions to get $-22 \leq x \leq 8$

Conclusion

In this article, we have solved the absolute value equation $|x+7| \leq 15$ using the two-case method. We have found the solution to be $-22 \leq x \leq 8$. This solution represents the range of values of $x$ that satisfy the equation.

Introduction

In our previous article, we explored the solution of the absolute value equation $|x+7| \leq 15$. We used the two-case method to find the solution, which was $-22 \leq x \leq 8$. In this article, we will answer some frequently asked questions about absolute value equations.

Q&A

Q1: What is an absolute value equation?

A1: An absolute value equation is a type of inequality that involves the absolute value of a variable or expression. It is used to represent the distance of a number from zero on the number line.

Q2: How do I solve an absolute value equation?

A2: To solve an absolute value equation, you need to consider two cases:

• Case 1: $x-a \geq 0$
• Case 2: $x-a < 0$

For case 1, the absolute value equation becomes $x-a \leq b$. For case 2, the absolute value equation becomes $-(x-a) \leq b$.

Q3: What is the two-case method for solving absolute value equations?

A3: The two-case method is a technique used to solve absolute value equations. It involves considering two cases: $x-a \geq 0$ and $x-a < 0$. For each case, you need to solve the equation separately.

Q4: How do I know which case to use?

A4: To determine which case to use, you need to check the sign of $x-a$. If $x-a \geq 0$, you use case 1. If $x-a < 0$, you use case 2.

Q5: What is the solution to the equation $|x+7| \leq 15$?

A5: The solution to the equation $|x+7| \leq 15$ is $-22 \leq x \leq 8$.

Q6: How do I combine the solutions from both cases?

A6: To combine the solutions from both cases, you need to find the intersection of the two solutions. In this case, the intersection is $-22 \leq x \leq 8$.

Q7: What is the final answer to the equation $|x+7| \leq 15$?

A7: The final answer to the equation $|x+7| \leq 15$ is $-22 \leq x \leq 8$.

Conclusion

In this article, we have answered some frequently asked questions about absolute value equations. We have explained the two-case method and how to combine the solutions from both cases. We have also provided the final answer to the equation $|x+7| \leq 15$.