Question

identify equivalent multistep expressions
which expression(s) is/are equivalent to
10 - 3(4x + 5).
explain or show work to justify your decision.

expression equivalent? (yes or no) explain
3(4x + 5)

10 - 12x - 15

10 - (12x + 15)

10 - 12x + 15

-12x - 5

Explanation:

Step1: Simplify the original expression

First, apply the distributive property to \(10 - 3(4x + 5)\). The distributive property states that \(a(b + c)=ab + ac\), so here \(a=-3\), \(b = 4x\), \(c = 5\). So we have:
\(10-3(4x + 5)=10-(3\times4x+3\times5)=10-(12x + 15)\)

Step2: Simplify further by distributing the negative sign

Distribute the negative sign inside the parentheses: \(10-(12x + 15)=10-12x-15\)

Step3: Combine like terms

Combine the constant terms \(10\) and \(-15\): \(10-12x - 15=-12x+(10 - 15)=-12x-5\)

Now we analyze each expression:

For \(3(4x + 5)\):

The original expression is \(10 - 3(4x + 5)\), which is \(10\) minus \(3(4x + 5)\), not equal to \(3(4x + 5)\). So equivalent? No.

For \(10 - 12x - 15\):

From Step 2, we saw that \(10 - 3(4x + 5)\) simplifies to \(10-12x - 15\). So equivalent? Yes.

For \(10-(12x + 15)\):

From Step 1, we have \(10 - 3(4x + 5)=10-(12x + 15)\). So equivalent? Yes.

For \(10 - 12x + 15\):

The original expression simplifies to \(10-12x - 15\), not \(10-12x + 15\) (the sign of \(15\) is wrong). So equivalent? No.

For \(-12x - 5\):

From Step 3, we have \(10 - 3(4x + 5)\) simplifies to \(-12x-5\). So equivalent? Yes.

Answer:

• \(3(4x + 5)\): No, because the original expression is \(10\) minus this expression, not equal to it.
• \(10 - 12x - 15\): Yes, by distributing \(-3\) in \(10 - 3(4x + 5)\) we get \(10-12x - 15\).
• \(10-(12x + 15)\): Yes, by applying the distributive property to \(10 - 3(4x + 5)\) we get \(10-(12x + 15)\).
• \(10 - 12x + 15\): No, the sign of \(15\) is incorrect after distribution.
• \(-12x - 5\): Yes, by combining like terms in \(10-12x - 15\) we get \(-12x-5\).