Question

multiple choice 1 point
gabriel is asked to show the steps for simplifying this
expression:
$3(2y + 5) - 4x \geq 7(2x + 1) + 4y$
his work is shown below.

1. $3(2y + 5) - 4x \geq 7(2x + 1) + 4y$
2. $6y + 15 - 4x \geq 14x + 7 + 4y$
3. $6y - 4y \geq 14x + 4x + 7 - 15$
4. $2y \geq 18x - 8$
5. $y \geq 9x - 4$

what property did gabriel apply between steps 1 and 2?
\bigcirc distributive property
\bigcirc associative property
\bigcirc additive identity
\bigcirc commutative property

Explanation:

To determine the property used between steps 1 and 2, we analyze the transformation. In step 1, we have \(3(2y + 5)-4x\) and \(7(2x + 1)+4y\). In step 2, these become \(6y + 15-4x\) and \(14x + 7+4y\). This is achieved by distributing the 3 into \(2y + 5\) (giving \(3\times2y+3\times5 = 6y + 15\)) and distributing the 7 into \(2x + 1\) (giving \(7\times2x+7\times1=14x + 7\)). The distributive property states that \(a(b + c)=ab+ac\), which matches this transformation. The associative property deals with grouping, additive identity is about adding 0, and commutative property is about order of addition/multiplication—none of these apply here.

Answer:

A. Distributive Property