Question

1. ms. carey wrote the equation $3x - 7 - \frac{2}{3}(9x - 6)$ on the board. write the simplified expression.

a. $-9x + 11$
b. $-3x + 3$
c. $-3x - 3$
d. $9x + 3$

Explanation:

Step1: Distribute the fraction

We have the expression \(3x - \frac{2}{3}(9x - 6)\). First, distribute \(\frac{2}{3}\) to both terms inside the parentheses:
\(\frac{2}{3} \times 9x = 6x\) and \(\frac{2}{3} \times (-6) = -4\)
So the expression becomes \(3x - 6x + 4\)

Step2: Combine like terms

Now, combine the \(x\)-terms: \(3x - 6x = -3x\)
Then, we have \(-3x + 4\)? Wait, no, wait. Wait, original expression is \(3x - \frac{2}{3}(9x - 6)\). Wait, maybe I misread. Wait, let's re - do:

Wait, the expression is \(3x-\frac{2}{3}(9x - 6)\)

Distribute \(-\frac{2}{3}\) (because it's minus \(\frac{2}{3}\) times the parentheses):

\(-\frac{2}{3}\times9x=-6x\) and \(-\frac{2}{3}\times(-6) = + 4\)

So now we have \(3x-6x + 4\)? Wait, no, the options don't have that. Wait, maybe the original expression is \(3x-7-\frac{2}{3}(9x - 6)\)? Wait, the image says "3x - 7 - 2/3(9x - 6)"? Wait, the user's image: "Ms. Carey wrote the equation 3x - 7 - 2/3(9x - 6) on the board. Write the simplified expression." Oh! I missed the -7. Let's start over.

Step1: Distribute the fraction

Given the expression \(3x-7-\frac{2}{3}(9x - 6)\). Distribute \(\frac{2}{3}\) to \(9x\) and \(-6\):
\(\frac{2}{3}\times9x = 6x\) and \(\frac{2}{3}\times(-6)=-4\)
So the expression becomes \(3x - 7-6x + 4\) (because we have \(-\frac{2}{3}(9x - 6)=-6x + 4\))

Step2: Combine like terms

Combine the \(x\)-terms: \(3x-6x=-3x\)
Combine the constant terms: \(-7 + 4=-3\)
So the simplified expression is \(-3x-3\)? Wait, no, the options are:

a. \(-9x + 11\)

b. \(-3x + 3\)

c. \(-3x-3\)

d. \(9x + 3\)

Wait, maybe the original expression is \(3x+7-\frac{2}{3}(9x - 6)\)? No, let's check the distribution again.

Wait, let's take the expression as \(3x-\frac{2}{3}(9x - 6)\) (without the -7, maybe the image had a typo). Let's do that:

Distribute \(-\frac{2}{3}\):

\(-\frac{2}{3}\times9x=-6x\), \(-\frac{2}{3}\times(-6) = 4\)

So \(3x-6x + 4=-3x + 4\). Not matching.

Wait, maybe the expression is \(3x+\frac{2}{3}(9x - 6)\)? No. Wait, let's check the options. Option c is \(-3x - 3\), option b is \(-3x + 3\).

Wait, let's re - examine the image. The image says: "Ms. Carey wrote the equation 3x - 7 - 2/3(9x - 6) on the board. Write the simplified expression."

So let's do that:

Expression: \(3x-7-\frac{2}{3}(9x - 6)\)

Step 1: Distribute \(-\frac{2}{3}\) to \(9x\) and \(-6\):

\(-\frac{2}{3}\times9x=-6x\)

\(-\frac{2}{3}\times(-6) = 4\)

So the expression becomes \(3x-7-6x + 4\)

Step 2: Combine like terms:

For \(x\) terms: \(3x-6x=-3x\)

For constant terms: \(-7 + 4=-3\)

So the simplified expression is \(-3x-3\), which is option c.

Wait, but let's check again. If we have \(3x-7-\frac{2}{3}(9x - 6)\)

\(=3x-7-6x + 4\) (because \(\frac{2}{3}\times9x = 6x\), \(\frac{2}{3}\times6 = 4\), and the negative sign in front of the fraction makes it \(-6x+4\))

Then \(3x-6x=-3x\) and \(-7 + 4=-3\), so \(-3x-3\), which is option c.

Answer:

c. \(-3x - 3\)