Properties for Equivalent Expressions - Pre Algebra

1. Which property justifies $3 + (x + 5) = (3 + x) + 5$?

   The associative property of addition. Only the grouping changes, not the order.

2. Which property justifies $x \cdot 7 = 7x$?

   The commutative property of multiplication. Only the order of the factors changes.

3. Use the distributive property to rewrite $4(2x - 3)$.

   $8x - 12$. Multiply 4 by each term: $4 \cdot 2x = 8x$ and $4 \cdot (-3) = -12$.

4. What does the commutative property of addition state?

   Changing the order of addends does not change the sum: $a + b = b + a$.

5. Factor out the greatest common factor of $6x + 9$.

   $3(2x + 3)$. The greatest common factor of 6 and 9 is 3.

6. Why does the commutative property not apply to subtraction?

   Changing the order changes the result: $5 - 2 = 3$, but $2 - 5 = -3$.

7. What does the distributive property state?

   Multiplying a sum by a number multiplies each addend: $a(b + c) = ab + ac$.

8. Why is $2(x + 5)$ not equivalent to $2x + 5$?

   The 2 must multiply both terms in the parentheses: $2(x + 5) = 2x + 10$.

9. Use the distributive property to rewrite $-3(x - 4)$.

   $-3x + 12$. Multiplying $-3$ by $-4$ gives positive 12.

10. Simplify $3(2x - 1) + 5(x + 2)$.

    $11x + 7$. Distribute to get $6x - 3 + 5x + 10$, then combine like terms.

11. Write $5(x + 2) - 2(x + 2)$ as a single product.

    $3(x + 2)$. Treat $(x + 2)$ as one quantity: $5 - 2 = 3$ copies of it.

12. What does the associative property of multiplication state?

    Changing the grouping of factors does not change the product: $(ab)c = a(bc)$.

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