Why These Statements Are True or False

Oct 13, 2025 by recoveryshake.com 81 views

In mathematics, inequalities are used to compare the values of different numbers or expressions. Given the statement $6 \ \textgreater \ -2$ , we need to determine which of the following statements are true. In this article, we will explore each statement and provide a detailed explanation of why it is or is not true.

Statement 1: $(2)(6) \ \textgreater \ (-2)(2)$

To evaluate this statement, we need to follow the order of operations (PEMDAS). First, we multiply 2 by 6, which gives us 12. Then, we multiply -2 by 2, which gives us -4. Since 12 is greater than -4, this statement is true.

Statement 2: $\frac{6}{2} \ \textless \ \frac{-2}{2}$

To evaluate this statement, we need to divide 6 by 2, which gives us 3. Then, we divide -2 by 2, which gives us -1. Since 3 is not less than -1, this statement is false.

Statement 3: $(-2)(6) \ \textgreater \ (-2)(-2)$

To evaluate this statement, we need to multiply -2 by 6, which gives us -12. Then, we multiply -2 by -2, which gives us 4. Since -12 is not greater than 4, this statement is false.

Why These Statements Are True or False

In the first statement, we multiplied 2 by 6 and -2 by 2, and we got 12 and -4, respectively. Since 12 is greater than -4, this statement is true. In the second statement, we divided 6 by 2 and -2 by 2, and we got 3 and -1, respectively. Since 3 is not less than -1, this statement is false. In the third statement, we multiplied -2 by 6 and -2 by -2, and we got -12 and 4, respectively. Since -12 is not greater than 4, this statement is false.

Conclusion

In conclusion, only the first statement is true. The other two statements are false. This is because the first statement follows the order of operations and the properties of inequalities, while the other two statements do not.

Key Takeaways

When evaluating inequalities, we need to follow the order of operations (PEMDAS).
We need to multiply and divide numbers correctly to get the correct result.
We need to understand the properties of inequalities, such as the commutative and associative properties.

Final Thoughts

In this article, we will answer some frequently asked questions related to inequalities and operations.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The order of operations is often remembered using the acronym PEMDAS, which stands for:

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

Q: What is the difference between > and <?

A: The symbol > is used to indicate that a value is greater than another value, while the symbol < is used to indicate that a value is less than another value. For example, 5 > 3 means that 5 is greater than 3, while 3 < 5 means that 3 is less than 5.

Q: Can I simplify an inequality by adding or subtracting the same value to both sides?

A: Yes, you can simplify an inequality by adding or subtracting the same value to both sides. This is known as "adding or subtracting the same value to both sides" or "isolating the variable." For example, if we have the inequality 2x + 3 > 5, we can subtract 3 from both sides to get 2x > 2.

Q: Can I multiply or divide both sides of an inequality by a negative number?

A: No, you cannot multiply or divide both sides of an inequality by a negative number. This is because multiplying or dividing by a negative number will change the direction of the inequality. For example, if we have the inequality x > 2, we cannot multiply both sides by -1 to get -x < -2, because this would change the direction of the inequality.

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

A: Yes, you can multiply or divide both sides of an inequality by a fraction, but you must be careful to multiply or divide both sides by the same fraction. For example, if we have the inequality x > 2, we can multiply both sides by 1/2 to get (1/2)x > 1.

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. This is known as "adding or subtracting the same value to both sides" or "isolating the variable." For example, if we have the inequality 2x + 3 > 5, we can subtract 3 from both sides to get 2x > 2.

Q: Can I use the distributive property to simplify an inequality?

A: Yes, you can use the distributive property to simplify an inequality. The distributive property states that for any numbers a, b, and c, a(b + c) = ab + ac. For example, if we have the inequality 2(x + 3) > 5, we can use the distributive property to simplify it to 2x + 6 > 5.

Conclusion

In this article, we answered some frequently asked questions related to inequalities and operations. We learned that we need to follow the order of operations, understand the properties of inequalities, and be careful when multiplying or dividing both sides of an inequality by a negative number or a fraction. By following these principles, we can confidently solve inequalities and operations.