Which Are Correct Representations Of The Inequality − 3 ( 2 X − 5 ) \textless 5 ( 2 − X -3(2x - 5) \ \textless \ 5(2 - X − 3 ( 2 X − 5 ) \textless 5 ( 2 − X ]? Select Two Options.A. X \textless 5 X \ \textless \ 5 X \textless 5 B. − 6 X − 5 \textless 10 − X -6x - 5 \ \textless \ 10 - X − 6 X − 5 \textless 10 − X C. − 6 X + 15 \textless 10 − 5 X -6x + 15 \ \textless \ 10 - 5x − 6 X + 15 \textless 10 − 5 X

In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems. An inequality is a statement that two expressions are not equal, and it can be represented in different forms, such as greater than, less than, greater than or equal to, and less than or equal to. In this article, we will focus on solving the inequality $-3(2x - 5) \ \textless \ 5(2 - x)$ and select the correct representations of the inequality.

Understanding the Given Inequality

The given inequality is $-3(2x - 5) \ \textless \ 5(2 - x)$. To solve this inequality, we need to follow the order of operations (PEMDAS) and simplify the expressions.

Step 1: Distribute the Negative 3

The first step is to distribute the negative 3 to the terms inside the parentheses.

-3(2x - 5) = -6x + 15

Step 2: Distribute the 5

Next, we need to distribute the 5 to the terms inside the parentheses.

5(2 - x) = 10 - 5x

Step 3: Rewrite the Inequality

Now that we have simplified the expressions, we can rewrite the inequality as:

-6x + 15 \  \textless \  10 - 5x

Selecting the Correct Representations

We are given three options to select the correct representations of the inequality. Let's analyze each option and determine which one is correct.

Option A: $x \ \textless \ 5$

To determine if this option is correct, we need to simplify the inequality and compare it to the given option.

-6x + 15 \  \textless \  10 - 5x

Subtracting 15 from both sides:

-6x \  \textless \  -5 - 5x

Adding 5x to both sides:

-x \  \textless \  -5

Multiplying both sides by -1:

x \  \textgreater \  5

This option is incorrect because it states that $x \ \textless \ 5$, whereas the correct solution is $x \ \textgreater \ 5$.

Option B: $-6x - 5 \ \textless \ 10 - x$

To determine if this option is correct, we need to simplify the inequality and compare it to the given option.

-6x + 15 \  \textless \  10 - 5x

Subtracting 15 from both sides:

-6x \  \textless \  -5 - 5x

Adding 5x to both sides:

-x \  \textless \  -5

Multiplying both sides by -1:

x \  \textgreater \  5

This option is incorrect because it states that $-6x - 5 \ \textless \ 10 - x$, whereas the correct solution is $x \ \textgreater \ 5$.

Option C: $-6x + 15 \ \textless \ 10 - 5x$

This option is correct because it represents the original inequality.

Conclusion

In conclusion, the correct representation of the inequality $-3(2x - 5) \ \textless \ 5(2 - x)$ is $-6x + 15 \ \textless \ 10 - 5x$. This option is the only correct representation of the inequality among the given options.

Final Answer

The final answer is:

• Option C: $-6x + 15 \ \textless \ 10 - 5x$
  Frequently Asked Questions (FAQs) About Solving Inequalities ====================================================================

In the previous article, we discussed how to solve the inequality $-3(2x - 5) \ \textless \ 5(2 - x)$ and select the correct representations of the inequality. In this article, we will answer some frequently asked questions (FAQs) about solving inequalities.

Q: What is an inequality?

A: An inequality is a statement that two expressions are not equal. It can be represented in different forms, such as greater than, less than, greater than or equal to, and less than or equal to.

Q: How do I solve an inequality?

A: To solve an inequality, you need to follow the order of operations (PEMDAS) and simplify the expressions. You can also use inverse operations to isolate the variable.

Q: What is the difference between an inequality and an equation?

A: An equation is a statement that two expressions are equal, whereas an inequality is a statement that two expressions are not equal.

Q: Can I add or subtract the same value to both sides of an inequality?

A: Yes, you can add or subtract the same value to both sides of an inequality. However, you cannot multiply or divide both sides of an inequality by a negative value.

Q: How do I multiply or divide both sides of an inequality by a negative value?

A: To multiply or divide both sides of an inequality by a negative value, you need to reverse the direction of the inequality sign.

Q: What is the concept of inverse operations?

A: Inverse operations are operations that "undo" each other. For example, addition and subtraction are inverse operations, as are multiplication and division.

Q: How do I use inverse operations to solve an inequality?

A: To use inverse operations to solve an inequality, you need to identify the inverse operation of the operation that is being performed on the variable. For example, if the inequality is $x + 3 \ \textless \ 5$, you can subtract 3 from both sides to isolate the variable.

Q: Can I use inverse operations to solve an inequality with fractions?

A: Yes, you can use inverse operations to solve an inequality with fractions. However, you need to be careful when multiplying or dividing both sides of the inequality by a fraction.

Q: How do I multiply or divide both sides of an inequality by a fraction?

A: To multiply or divide both sides of an inequality by a fraction, you need to multiply or divide both sides of the inequality by the reciprocal of the fraction.

Q: What is the concept of the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that dictate the order in which mathematical operations should be performed. The acronym PEMDAS stands for:

• Parentheses
• Exponents
• Multiplication and Division
• Addition and Subtraction

Q: How do I apply the order of operations (PEMDAS) to solve an inequality?

A: To apply the order of operations (PEMDAS) to solve an inequality, you need to follow the order of operations and simplify the expressions.

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

A: Yes, you can use a calculator to solve an inequality. However, you need to be careful when using a calculator to solve an inequality, as it may not always give you the correct solution.

Conclusion

In conclusion, solving inequalities requires a clear understanding of the concept of inequalities, the order of operations (PEMDAS), and inverse operations. By following the steps outlined in this article, you can solve inequalities with confidence.

Final Answer

The final answer is:

• Yes, you can solve inequalities using a calculator, but be careful when using it.
• The order of operations (PEMDAS) is a set of rules that dictate the order in which mathematical operations should be performed.
• Inverse operations are operations that "undo" each other.
• You can use inverse operations to solve an inequality with fractions, but be careful when multiplying or dividing both sides of the inequality by a fraction.