Final answer:
The equivalent expressions are A: 2.5 + 4(-1.5 + 10) + 3, C: 2.5 - 4(10 - 1.5) + 3, D: 3 + 2.5 - 4(1.5 - 10), and E: -4(1.5 - 10) + 3 + 2.5, as they respect the commutative and distributive properties in mathematics.
Step-by-step explanation:
The expressions equivalent to 2.5 - 4(1.5 - 10) + 3 can be identified by applying the distributive property and the commutative property of addition. It's important to note that the sign of the number changes when subtracting it, similar to the rule shown in 5 - (+3) = 5 - 3 = 2. Now, let's analyze the given options:
- A 2.5 + 4(-1.5 + 10) + 3 is equivalent because the distributive property is applied correctly to change the sign within the parentheses.
- B 2.5 - 4 + 3(1.5 - 10) is not equivalent because the multiplication by 4 is applied incorrectly.
- C 2.5 - 4(10 - 1.5) + 3 is equivalent. The subtraction inside the brackets is reversed, but due to the associative property of multiplication, the overall expression remains equivalent.
- D 3 + 2.5 - 4(1.5 - 10) is equivalent because it's just the original expression with the terms 3 and 2.5 reordered due to the commutative property of addition.
- E -4(1.5 - 10) + 3 + 2.5 is equivalent because it is the original expression with the terms reordered due to the commutative property.
- F 4 - 2.5(1.5 - 10) + 3 is not equivalent because the terms from the original expression have been altered improperly.
Therefore, the equivalent expressions are A, C, D, and E.