Question

for problems 1 - 4, distribute and combine like terms to write an equivalent expression. show your work.

1. $9 + 3(2x + 4)$
2. $\frac{1}{4}(8x + 40) + 6$
3. $2x + 5(4x - 5)$
4. $8(x + 3) + 5(2x + 6)$
5. which expressions are equivalent to $8x + \frac{1}{3}(6x - 12)$? choose all that apply.

a. $8x + 2x - 12$
b. $8x + \frac{1}{2}x - \frac{1}{12}$
c. $8x + 2x - 4$
d. $10x - 4$
e. $8x + 2(x - 2)$
f. $8x + 3(x - 4)$

1. kendra says that the expressions $6 + 3(3 - x)$ and $31 - 3x$ are equivalent because they evaluate to the same number when $x = 0$. is kendra correct? explain.

Explanation:

Step1: Simplify the original expression

We start with the expression \(8x+\frac{1}{3}(6x - 12)\). First, we distribute the \(\frac{1}{3}\) to both terms inside the parentheses:
\(\frac{1}{3}\times6x=2x\) and \(\frac{1}{3}\times(- 12)=- 4\)
So the expression becomes \(8x + 2x-4\)

Step2: Combine like terms

Now we combine the \(x\) terms: \(8x+2x = 10x\)
So the simplified expression is \(10x - 4\)

Step3: Analyze option A

Option A is \(8x + 2x-12\). The constant term here is \(- 12\) while in our simplified expression it is \(-4\), so option A is incorrect.

Step4: Analyze option B

Option B is \(8x+\frac{1}{3}x-\frac{1}{12}\). The coefficients of \(x\) and the constant term do not match our simplified expression, so option B is incorrect.

Step5: Analyze option C

Option C is \(8x + 2x-4\). This is the expression we got after distributing the \(\frac{1}{3}\) and before combining like terms completely, but it is equivalent to the original expression (just a step in the simplification process).

Step6: Analyze option D

Option D is \(10x-4\). This is the fully simplified form of the original expression, so it is equivalent.

Step7: Analyze option E

Option E is \(8x + 2(x - 2)\). Let's simplify this:
First, distribute the \(2\): \(8x+2x-4\)
Then combine like terms: \(10x - 4\), which is equivalent to the original expression.

Step8: Analyze option F

Option F is \(8x+3(x - 4)\). Distribute the \(3\): \(8x + 3x-12=11x-12\), which is not equivalent to the original expression.

Answer:

C. \(8x + 2x-4\), D. \(10x - 4\), E. \(8x + 2(x - 2)\)