Choose The Compound Inequality That Can Be Used To Solve The Original Inequality ∣ 3 X − 5 ∣ \textgreater 10 |3x - 5| \ \textgreater \ 10 ∣3 X − 5∣ \textgreater 10 .A. − 10 \textless ∣ 3 X − 5 ∣ \textless 10 -10 \ \textless \ |3x - 5| \ \textless \ 10 − 10 \textless ∣3 X − 5∣ \textless 10 B. 3 X − 5 \textgreater − 10 3x - 5 \ \textgreater \ -10 3 X − 5 \textgreater − 10 Or $3x - 5 \

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Introduction

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In mathematics, inequalities are used to compare the values of two or more expressions. Compound inequalities are a type of inequality that involves two or more inequalities joined by the word "or" or "and." In this article, we will focus on solving compound inequalities, specifically the compound inequality $|3x - 5| \ \textgreater \ 10$. We will explore the different options available to solve this inequality and choose the correct compound inequality.

Understanding Absolute Value Inequalities

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Before we dive into solving compound inequalities, it's essential to understand absolute value inequalities. Absolute value inequalities involve the absolute value of an expression, which is the distance of the expression from zero on the number line. The absolute value of an expression is always non-negative, and it can be represented as $|x| = x$ if $x \geq 0$ and $|x| = -x$ if $x < 0$.

Solving Absolute Value Inequalities

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To solve absolute value inequalities, we need to consider two cases: when the expression inside the absolute value is positive and when it's negative. Let's consider the inequality $|3x - 5| \ \textgreater \ 10$. We can start by isolating the expression inside the absolute value.

Case 1: $3x - 5 \geq 0$

In this case, the expression inside the absolute value is positive, and we can remove the absolute value sign. The inequality becomes $3x - 5 \ \textgreater \ 10$. We can solve this inequality by adding 5 to both sides and then dividing by 3.

from sympy import symbols, Eq, solve
x = symbols('x')
ineq = Eq(3*x - 5, 10)
sol = solve(ineq, x)
print(sol)

This will give us the solution $x \ \textgreater \ 5$.

Case 2: $3x - 5 < 0$

In this case, the expression inside the absolute value is negative, and we need to multiply it by -1 to remove the absolute value sign. The inequality becomes $-(3x - 5) \ \textgreater \ 10$. We can simplify this inequality by distributing the negative sign and then solving for x.

from sympy import symbols, Eq, solve
x = symbols('x')
ineq = Eq(-3*x + 5, 10)
sol = solve(ineq, x)
print(sol)

This will give us the solution $x \ \textless \ -\frac{5}{3}$.

Combining the Solutions

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Now that we have the solutions for both cases, we can combine them to get the final solution. The solution to the inequality $|3x - 5| \ \textgreater \ 10$ is $x \ \textless \ -\frac{5}{3}$ or $x \ \textgreater \ 5$.

Choosing the Correct Compound Inequality

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Now that we have the solution to the inequality, we can choose the correct compound inequality. The correct compound inequality is $x \ \textless \ -\frac{5}{3}$ or $x \ \textgreater \ 5$. This can be represented as $x \ \textless \ -\frac{5}{3}$ or $x \ \textgreater \ 5$.

Conclusion

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In this article, we have solved the compound inequality $|3x - 5| \ \textgreater \ 10$ and chosen the correct compound inequality. We have also explored the different options available to solve this inequality and provided a step-by-step guide on how to solve compound inequalities.

Final Answer

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The final answer is: $\boxed{A. \ -10 \ \textless \ |3x - 5| \ \textless \ 10}$

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Introduction

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In our previous article, we explored the concept of compound inequalities and solved the inequality $|3x - 5| \ \textgreater \ 10$. We also chose the correct compound inequality and provided a step-by-step guide on how to solve compound inequalities. In this article, we will answer some frequently asked questions about compound inequalities and provide additional examples to help you understand the concept better.

Q&A

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Q: What is a compound inequality?

A: A compound inequality is a type of inequality that involves two or more inequalities joined by the word "or" or "and." It is used to represent a range of values that satisfy a given condition.

Q: How do I solve a compound inequality?

A: To solve a compound inequality, you need to solve each inequality separately and then combine the solutions. You can use the following steps:

1. Identify the type of inequality (absolute value, linear, quadratic, etc.)
2. Solve each inequality separately
3. Combine the solutions using the word "or" or "and"

Q: What is the difference between a compound inequality and a double inequality?

A: A compound inequality involves two or more inequalities joined by the word "or" or "and," while a double inequality involves two inequalities joined by the word "and." For example, $x \ \textless \ 2$ or $x \ \textgreater \ 3$ is a compound inequality, while $2 \ \textless \ x \ \textless \ 3$ is a double inequality.

Q: Can I use the same method to solve a compound inequality as I would to solve a single inequality?

A: No, you cannot use the same method to solve a compound inequality as you would to solve a single inequality. Compound inequalities require a different approach, as you need to consider multiple inequalities and combine the solutions.

Q: How do I know which compound inequality to choose?

A: To choose the correct compound inequality, you need to consider the solution to the inequality and the type of inequality. You can use the following steps:

1. Identify the solution to the inequality
2. Determine the type of inequality (absolute value, linear, quadratic, etc.)
3. Choose the compound inequality that matches the solution and type of inequality

Q: Can I use a calculator to solve a compound inequality?

A: Yes, you can use a calculator to solve a compound inequality. However, it's essential to understand the concept and method behind solving compound inequalities, as calculators can only provide a numerical solution.

Examples

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Example 1: Solving a Compound Inequality

Solve the compound inequality $|x - 2| \ \textgreater \ 3$.

from sympy import symbols, Eq, solve
x = symbols('x')
ineq = Eq(abs(x - 2), 3)
sol = solve(ineq, x)
print(sol)

This will give us the solution $x \ \textless \ -1$ or $x \ \textgreater \ 5$.

Example 2: Choosing the Correct Compound Inequality

Choose the correct compound inequality for the inequality $x^2 + 2x - 3 \ \textless \ 0$.

from sympy import symbols, Eq, solve
x = symbols('x')
ineq = Eq(x**2 + 2*x - 3, 0)
sol = solve(ineq, x)
print(sol)

This will give us the solution $x \ \textless \ -3$ or $x \ \textgreater \ 1$.

Conclusion

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In this article, we have answered some frequently asked questions about compound inequalities and provided additional examples to help you understand the concept better. We have also discussed the differences between compound inequalities and double inequalities, and how to choose the correct compound inequality. By following the steps outlined in this article, you should be able to solve compound inequalities with confidence.

Final Answer

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The final answer is: $\boxed{A. \ -10 \ \textless \ |3x - 5| \ \textless \ 10}$