Question

determine which of the expressions in the table is equivalent to: -2.4a - 9.2b + 7.5
explain or show work to justify your decisions.

expression	equivalent? (yes or no)	explain
b -2.4(a + 9.2b - 7.5)
c 1/3(-2.4a - 9.2b + 7.5)·3
d -2.4a - 9.2b + 0·6c + 7.5
e -(2.4a - 9.2b + 7.5)

Explanation:

Step1: Analyze Expression A

Simplify \( -2.4a - 9.2b+(7.5 + 4.1)-4.1 \). First, calculate \( (7.5 + 4.1)-4.1=7.5+4.1 - 4.1 = 7.5 \). So the expression becomes \( -2.4a - 9.2b+7.5 \), which matches the original. So A is equivalent.

Step2: Analyze Expression B

Expand \( -2.4(a + 9.2b-7.5) \) using the distributive property: \( -2.4a-2.4\times9.2b + 2.4\times7.5 \). \( -2.4\times9.2b=-22.08b
eq - 9.2b \), so B is not equivalent.

Step3: Analyze Expression C

Simplify \( \frac{1}{3}(-2.4a - 9.2b + 7.5)\cdot3 \). The \( \frac{1}{3} \) and \( 3 \) cancel out, leaving \( -2.4a - 9.2b + 7.5 \), which matches the original. So C is equivalent.

Step4: Analyze Expression D

The original expression has no \( c \) term, but D has \( 0.6c \). Unless \( c = 0 \), it's not equivalent. So D is not equivalent (since \( c \) is a variable, not necessarily 0).

Step5: Analyze Expression E

Expand \( -(2.4a - 9.2b + 7.5)=-2.4a + 9.2b-7.5 \). The sign of \( b \) and the constant term differ from the original, so E is not equivalent.

Answer:

A. \( -2.4a - 9.2b+(7.5 + 4.1)-4.1 \) (yes), C. \( \frac{1}{3}(-2.4a - 9.2b + 7.5)\cdot3 \) (yes)