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. In this article, we will focus on solving the inequality $-3(2x-5)\ \textless \ 5(2-x)$ and provide two correct representations of the solution.

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 Sign

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

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

Step 2: Distribute the Positive Sign

Next, we distribute the positive sign to the terms inside the parentheses.

$$5(2-x) = 10 - 5x$$

Step 3: Rewrite the Inequality

Now, we can rewrite the inequality using the simplified expressions.

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

Solving the Inequality

To solve the inequality, we need to isolate the variable $x$ on one side of the inequality.

Step 4: Add $5x$ to Both Sides

We add $5x$ to both sides of the inequality to get rid of the negative term.

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

Step 5: Simplify the Inequality

Now, we simplify the inequality by combining like terms.

$$-x + 15\ \textless \ 10$$

Step 6: Subtract 15 from Both Sides

We subtract 15 from both sides of the inequality to isolate the term with the variable.

$$-x + 15 - 15\ \textless \ 10 - 15$$

Step 7: Simplify the Inequality

Now, we simplify the inequality by combining like terms.

$$-x\ \textless \ -5$$

Step 8: Multiply Both Sides by -1

To solve for $x$, we multiply both sides of the inequality by -1. Remember to reverse the direction of the inequality when multiplying by a negative number.

$$x\ \textgreater \ 5$$

Correct Representations of the Inequality

Now that we have solved the inequality, we can provide two correct representations of the solution.

Option A: $x\ \textless \ 5$

This option is incorrect because the solution to the inequality is $x\ \textgreater \ 5$, not $x\ \textless \ 5$.

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

This option is incorrect because the solution to the inequality is $x\ \textgreater \ 5$, not $-6x-5\ \textless \ 10-x$.

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

This option is correct because it represents the solution to the inequality $-6x+15\ \textless \ 10-5x$.

Conclusion

In conclusion, the correct representations of the inequality $-3(2x-5)\ \textless \ 5(2-x)$ are $x\ \textgreater \ 5$ and $-6x+15\ \textless \ 10-5x$. We hope this article has provided a clear understanding of how to solve inequalities and provide correct representations of the solution.

Additional Tips and Resources

• When solving inequalities, always follow the order of operations (PEMDAS) and simplify the expressions.
• When multiplying or dividing both sides of an inequality by a negative number, remember to reverse the direction of the inequality.
• Practice solving inequalities with different variables and coefficients to become more comfortable with the process.

References

• [1] Khan Academy. (n.d.). Solving Inequalities. Retrieved from https://www.khanacademy.org/math/algebra/x2f1c6d7/x2f1c6d7/x2f1c6d7
• [2] Mathway. (n.d.). Solving Inequalities. Retrieved from https://www.mathway.com/subjects/inequalities

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

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

A: An inequality is a statement that two expressions are not equal, while an equation is a statement that two expressions are equal. Inequalities are often used to represent relationships between variables that are not equal.

Q: How do I solve an inequality with a variable on both sides?

A: To solve an inequality with a variable on both sides, you need to isolate the variable on one side of the inequality. This can be done by adding or subtracting the same value to both sides of the inequality, or by multiplying or dividing both sides by the same non-zero value.

Q: What is the order of operations (PEMDAS) and how does it apply to solving inequalities?

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:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

When solving inequalities, it is essential to follow the order of operations to ensure that the correct solution is obtained.

Q: How do I handle negative numbers when solving inequalities?

A: When solving inequalities, negative numbers can be handled in the same way as positive numbers. However, when multiplying or dividing both sides of an inequality by a negative number, remember to reverse the direction of the inequality.

Q: Can I use the same methods to solve linear inequalities as I would to solve linear equations?

A: Yes, the same methods can be used to solve linear inequalities as linear equations. However, when solving linear inequalities, it is essential to remember to reverse the direction of the inequality when multiplying or dividing both sides by a negative number.

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

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

• Not following the order of operations (PEMDAS)
• Not isolating the variable on one side of the inequality
• Not reversing the direction of the inequality when multiplying or dividing both sides by a negative number
• Not checking the solution to ensure that it satisfies the original inequality

Q: How can I practice solving inequalities?

A: There are many resources available to practice solving inequalities, including:

• Online resources such as Khan Academy and Mathway
• Practice problems in algebra textbooks
• Online practice tests and quizzes
• Working with a tutor or teacher to practice solving inequalities

Conclusion

In conclusion, solving inequalities requires a clear understanding of the order of operations (PEMDAS) and the ability to isolate the variable on one side of the inequality. By following the steps outlined in this article and practicing regularly, you can become more confident in your ability to solve inequalities.

Additional Tips and Resources

• When solving inequalities, always follow the order of operations (PEMDAS) and simplify the expressions.
• When multiplying or dividing both sides of an inequality by a negative number, remember to reverse the direction of the inequality.
• Practice solving inequalities with different variables and coefficients to become more comfortable with the process.
• Use online resources such as Khan Academy and Mathway to practice solving inequalities.

References

• [1] Khan Academy. (n.d.). Solving Inequalities. Retrieved from https://www.khanacademy.org/math/algebra/x2f1c6d7/x2f1c6d7/x2f1c6d7
• [2] Mathway. (n.d.). Solving Inequalities. Retrieved from https://www.mathway.com/subjects/inequalities