Which Inequality Is Equivalent To $|x-4|\ \textless \ 9$?A. $-9 \ \textgreater \ X-4 \ \textless \ 9$ B. $-9 \ \textless \ X-4 \ \textless \ 9$ C. $x-4 \ \textless \ -9$ Or $x-4 \ \textless \

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Understanding Absolute Value Inequalities

Absolute value inequalities are a type of mathematical expression that involves the absolute value of a variable or expression. In this case, we are given the inequality $|x-4|\ \textless \ 9$, and we need to determine which of the given options is equivalent to it.

The Basics of Absolute Value

Before we dive into the solution, let's quickly review the basics of absolute value. The absolute value of a number $a$, denoted by $|a|$, is the distance of $a$ from zero on the number line. In other words, it is the magnitude of $a$ without considering its direction. For example, the absolute value of $-3$ is $3$, and the absolute value of $3$ is also $3$.

The Given Inequality

The given inequality is $|x-4|\ \textless \ 9$. This means that the distance of $x-4$ from zero is less than $9$. In other words, $x-4$ can be anywhere between $-9$ and $9$, but not including $-9$ and $9$.

Option A: $-9 \ \textgreater \ x-4 \ \textless \ 9$

Let's analyze option A: $-9 \ \textgreater \ x-4 \ \textless \ 9$. This inequality states that $x-4$ is greater than $-9$ and less than $9$. However, this is not equivalent to the given inequality $|x-4|\ \textless \ 9$. To see why, let's consider the number line. If $x-4$ is greater than $-9$, it means that $x-4$ can be anywhere between $-9$ and infinity. Similarly, if $x-4$ is less than $9$, it means that $x-4$ can be anywhere between negative infinity and $9$. However, the given inequality $|x-4|\ \textless \ 9$ only allows $x-4$ to be anywhere between $-9$ and $9$, but not including $-9$ and $9$.

Option B: $-9 \ \textless \ x-4 \ \textless \ 9$

Now, let's analyze option B: $-9 \ \textless \ x-4 \ \textless \ 9$. This inequality states that $x-4$ is greater than $-9$ and less than $9$. This is equivalent to the given inequality $|x-4|\ \textless \ 9$. To see why, let's consider the number line. If $x-4$ is greater than $-9$ and less than $9$, it means that $x-4$ can be anywhere between $-9$ and $9$, but not including $-9$ and $9$. This is exactly what the given inequality $|x-4|\ \textless \ 9$ states.

Option C: $x-4 \ \textless \ -9$ or $x-4 \ \textless \ 9$

Finally, let's analyze option C: $x-4 \ \textless \ -9$ or $x-4 \ \textless \ 9$. This inequality states that $x-4$ is less than $-9$ or less than $9$. However, this is not equivalent to the given inequality $|x-4|\ \textless \ 9$. To see why, let's consider the number line. If $x-4$ is less than $-9$, it means that $x-4$ can be anywhere between negative infinity and $-9$. Similarly, if $x-4$ is less than $9$, it means that $x-4$ can be anywhere between negative infinity and $9$. However, the given inequality $|x-4|\ \textless \ 9$ only allows $x-4$ to be anywhere between $-9$ and $9$, but not including $-9$ and $9$.

Conclusion

In conclusion, the correct answer is option B: $-9 \ \textless \ x-4 \ \textless \ 9$. This inequality is equivalent to the given inequality $|x-4|\ \textless \ 9$. The other options are not equivalent to the given inequality.

Frequently Asked Questions

• What is the absolute value of a number? The absolute value of a number $a$, denoted by $|a|$, is the distance of $a$ from zero on the number line.
• What is the given inequality? The given inequality is $|x-4|\ \textless \ 9$.
• What is option A? Option A is $-9 \ \textgreater \ x-4 \ \textless \ 9$.
• What is option B? Option B is $-9 \ \textless \ x-4 \ \textless \ 9$.
• What is option C? Option C is $x-4 \ \textless \ -9$ or $x-4 \ \textless \ 9$.

Final Answer

The final answer is option B: $-9 \ \textless \ x-4 \ \textless \ 9$.

Understanding Absolute Value Inequalities

Absolute value inequalities are a type of mathematical expression that involves the absolute value of a variable or expression. In this article, we will explore some common questions and answers related to absolute value inequalities.

Q: What is the absolute value of a number?

A: The absolute value of a number $a$, denoted by $|a|$, is the distance of $a$ from zero on the number line.

Q: What is the given inequality?

A: The given inequality is $|x-4|\ \textless \ 9$.

Q: What is the meaning of the given inequality?

A: The given inequality means that the distance of $x-4$ from zero is less than $9$. In other words, $x-4$ can be anywhere between $-9$ and $9$, but not including $-9$ and $9$.

Q: What is option A?

A: Option A is $-9 \ \textgreater \ x-4 \ \textless \ 9$.

Q: What is option B?

A: Option B is $-9 \ \textless \ x-4 \ \textless \ 9$.

Q: What is option C?

A: Option C is $x-4 \ \textless \ -9$ or $x-4 \ \textless \ 9$.

Q: Which option is equivalent to the given inequality?

A: Option B is equivalent to the given inequality $|x-4|\ \textless \ 9$.

Q: Why is option A not equivalent to the given inequality?

A: Option A is not equivalent to the given inequality because it allows $x-4$ to be anywhere between $-9$ and infinity, and also between negative infinity and $9$, which is not what the given inequality states.

Q: Why is option C not equivalent to the given inequality?

A: Option C is not equivalent to the given inequality because it allows $x-4$ to be anywhere between negative infinity and $-9$, and also between negative infinity and $9$, which is not what the given inequality states.

Q: How do I solve absolute value inequalities?

A: To solve absolute value inequalities, you need to consider two cases: when the expression inside the absolute value is positive, and when it is negative. You then need to solve the resulting inequalities separately.

Q: What are some common mistakes to avoid when solving absolute value inequalities?

A: Some common mistakes to avoid when solving absolute value inequalities include:

• Not considering both cases when the expression inside the absolute value is positive and negative.
• Not solving the resulting inequalities separately.
• Not including the endpoints of the solution set.

Q: How do I graph absolute value inequalities?

A: To graph absolute value inequalities, you need to graph the expression inside the absolute value on the number line, and then shade the region that satisfies the inequality.

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

A: Absolute value inequalities have many real-world applications, including:

• Modeling distance and time relationships.
• Modeling financial transactions.
• Modeling physical phenomena such as temperature and pressure.

Q: How do I use absolute value inequalities in real-world problems?

A: To use absolute value inequalities in real-world problems, you need to:

• Identify the variables and constants in the problem.
• Write an absolute value inequality that models the problem.
• Solve the inequality to find the solution set.
• Interpret the solution set in the context of the problem.

Q: What are some common types of absolute value inequalities?

A: Some common types of absolute value inequalities include:

• Linear absolute value inequalities.
• Quadratic absolute value inequalities.
• Polynomial absolute value inequalities.

Q: How do I solve linear absolute value inequalities?

A: To solve linear absolute value inequalities, you need to:

• Write the inequality in the form $|ax+b|\ \textless \ c$.
• Solve the inequality by considering two cases: when $ax+b$ is positive, and when it is negative.
• Solve the resulting inequalities separately.

Q: How do I solve quadratic absolute value inequalities?

A: To solve quadratic absolute value inequalities, you need to:

• Write the inequality in the form $|ax^2+bx+c|\ \textless \ d$.
• Solve the inequality by considering two cases: when $ax^2+bx+c$ is positive, and when it is negative.
• Solve the resulting inequalities separately.

Q: How do I solve polynomial absolute value inequalities?

A: To solve polynomial absolute value inequalities, you need to:

• Write the inequality in the form $|a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0|\ \textless \ b$.
• Solve the inequality by considering two cases: when $a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ is positive, and when it is negative.
• Solve the resulting inequalities separately.

Q: What are some common mistakes to avoid when solving absolute value inequalities?

A: Some common mistakes to avoid when solving absolute value inequalities include:

• Not considering both cases when the expression inside the absolute value is positive and negative.
• Not solving the resulting inequalities separately.
• Not including the endpoints of the solution set.

Q: How do I graph absolute value inequalities?

A: To graph absolute value inequalities, you need to graph the expression inside the absolute value on the number line, and then shade the region that satisfies the inequality.

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

A: Absolute value inequalities have many real-world applications, including:

• Modeling distance and time relationships.
• Modeling financial transactions.
• Modeling physical phenomena such as temperature and pressure.

Q: How do I use absolute value inequalities in real-world problems?

A: To use absolute value inequalities in real-world problems, you need to:

• Identify the variables and constants in the problem.
• Write an absolute value inequality that models the problem.
• Solve the inequality to find the solution set.
• Interpret the solution set in the context of the problem.

Q: What are some common types of absolute value inequalities?

A: Some common types of absolute value inequalities include:

• Linear absolute value inequalities.
• Quadratic absolute value inequalities.
• Polynomial absolute value inequalities.

Q: How do I solve linear absolute value inequalities?

A: To solve linear absolute value inequalities, you need to:

• Write the inequality in the form $|ax+b|\ \textless \ c$.
• Solve the inequality by considering two cases: when $ax+b$ is positive, and when it is negative.
• Solve the resulting inequalities separately.

Q: How do I solve quadratic absolute value inequalities?

A: To solve quadratic absolute value inequalities, you need to:

• Write the inequality in the form $|ax^2+bx+c|\ \textless \ d$.
• Solve the inequality by considering two cases: when $ax^2+bx+c$ is positive, and when it is negative.
• Solve the resulting inequalities separately.

Q: How do I solve polynomial absolute value inequalities?

A: To solve polynomial absolute value inequalities, you need to:

• Write the inequality in the form $|a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0|\ \textless \ b$.
• Solve the inequality by considering two cases: when $a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ is positive, and when it is negative.
• Solve the resulting inequalities separately.

Q: What are some common mistakes to avoid when solving absolute value inequalities?

A: Some common mistakes to avoid when solving absolute value inequalities include:

• Not considering both cases when the expression inside the absolute value is positive and negative.
• Not solving the resulting inequalities separately.
• Not including the endpoints of the solution set.

Q: How do I graph absolute value inequalities?

A: To graph absolute value inequalities, you need to graph the expression inside the absolute value on the number line, and then shade the region that satisfies the inequality.

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

A: Absolute value inequalities have many real-world applications, including:

• Modeling distance and time relationships.
• Modeling financial transactions.
• Modeling physical phenomena such as temperature and pressure.

Q: How do I use absolute value inequalities in real-world problems?

A: To use absolute value inequalities in real-world problems, you need to:

• Identify the variables and constants in the problem.
• Write an absolute value inequality that models the problem.
• Solve the inequality to find the solution set.
• Interpret the solution set in the context of the problem.

Q: What are some common types of absolute value inequalities?

A: Some common types of absolute value inequalities include:

• Linear absolute value inequalities.
• Quadratic absolute value inequalities.
• Polynomial absolute value inequalities.

Q: How do I solve linear absolute value inequalities?

A: To solve linear absolute value inequalities, you need to:

• Write the inequality in the form $|ax+b|\ \textless \ c$.
• Solve the inequality by considering two cases: when $ax+b$ is positive, and when it is negative.
• Solve the resulting inequalities separately.

Q: How do I solve quadratic absolute value inequalities?

A: To solve quadratic absolute value inequalities, you need to:

• Write the inequality in the form $|ax^2+bx+c|\ \textless \ d$.