Select all statements that are true. Be prepared to explain your reasoning.

Select all statements that are true. Be prepared to explain your reasoning.

1. 4-2(3+7)=4-2•3-2•7
2. 4-2(3+7)=4+-2•3+-2•7
3. 4-2(3+7)=4-2•3+2•7
4. 4-2(3+7)= 4-(2•3+2•7)

The Correct Answer and Explanation is:

To determine which statements are true, we must first evaluate the left side of the equation ( 4 – 2(3 + 7) ) and then analyze each statement to see if they correctly represent the expression.

Step 1: Evaluate the Left Side

The expression is:
[
4 – 2(3 + 7)
]
First, we evaluate the expression inside the parentheses:
[
3 + 7 = 10
]
Now substitute back into the equation:
[
4 – 2(10)
]
Now, perform the multiplication:
[
4 – 20
]
Finally, calculate:
[
4 – 20 = -16
]

Step 2: Analyze Each Statement

1. Statement 1: ( 4 – 2(3 + 7) = 4 – 2 \cdot 3 – 2 \cdot 7 )
   Left Side: ( -16 )
   Right Side:
   [
   4 – 2 \cdot 3 – 2 \cdot 7 = 4 – 6 – 14 = 4 – 20 = -16
   ]
   Conclusion: True
2. Statement 2: ( 4 – 2(3 + 7) = 4 + -2 \cdot 3 + -2 \cdot 7 )
   Left Side: ( -16 )
   Right Side:
   [
   4 + -2 \cdot 3 + -2 \cdot 7 = 4 – 6 – 14 = 4 – 20 = -16
   ]
   Conclusion: True
3. Statement 3: ( 4 – 2(3 + 7) = 4 – 2 \cdot 3 + 2 \cdot 7 )
   Left Side: ( -16 )
   Right Side:
   [
   4 – 2 \cdot 3 + 2 \cdot 7 = 4 – 6 + 14 = 4 + 8 = 12
   ]
   Conclusion: False
4. Statement 4: ( 4 – 2(3 + 7) = 4 – (2 \cdot 3 + 2 \cdot 7) )
   Left Side: ( -16 )
   Right Side:
   [
   4 – (2 \cdot 3 + 2 \cdot 7) = 4 – (6 + 14) = 4 – 20 = -16
   ]
   Conclusion: True

Final Summary

The true statements are 1, 2, and 4.

Explanation

The key to solving these expressions lies in understanding the order of operations (parentheses, exponents, multiplication and division (left to right), addition and subtraction (left to right), often abbreviated as PEMDAS). In the first two statements, the manipulation of terms correctly maintains the structure of the original expression. In contrast, the third statement alters the addition and subtraction arrangement incorrectly, leading to a different value. The fourth statement correctly employs the distributive property, aligning with the value of the left side. By adhering to these mathematical principles, we can confidently analyze the truth of each statement.