*x*−3(32​*x*+3)=

3(−23​*x*+1)−23​*x*

Let's solve the equation:

\[
x - 3\left( \frac{3}{2}x + 3 \right) = 3\left( -\frac{2}{3}x + 1 \right) - \frac{2}{3}x
\]

### Step 1: Expand both sides

Expand the left-hand side:

\[
x - 3\left( \frac{3}{2}x + 3 \right) = x - \left( \frac{9}{2}x + 9 \right)
\]
\[
= x - \frac{9}{2}x - 9
\]

Now expand the right-hand side:

\[
3\left( -\frac{2}{3}x + 1 \right) - \frac{2}{3}x = -2x + 3 - \frac{2}{3}x
\]

### Step 2: Simplify both sides

For the left-hand side:

\[
x - \frac{9}{2}x - 9 = \frac{2}{2}x - \frac{9}{2}x - 9 = -\frac{7}{2}x - 9
\]

For the right-hand side:

\[
-2x - \frac{2}{3}x + 3
\]

We need a common denominator to combine the terms with \(x\):

\[
-2x - \frac{2}{3}x = -\frac{6}{3}x - \frac{2}{3}x = -\frac{8}{3}x
\]

Thus, the right-hand side becomes:

\[
-\frac{8}{3}x + 3
\]

### Step 3: Set the two expressions equal

Now, we have:

\[
-\frac{7}{2}x - 9 = -\frac{8}{3}x + 3
\]

### Step 4: Eliminate the fractions

Multiply the entire equation by 6 to eliminate the denominators:

\[
6\left( -\frac{7}{2}x - 9 \right) = 6\left( -\frac{8}{3}x + 3 \right)
\]

This simplifies to:

\[
-21x - 54 = -16x + 18
\]

### Step 5: Solve for \(x\)

Move all the terms with \(x\) to one side and constants to the other:

\[
-21x + 16x = 18 + 54
\]
\[
-5x = 72
\]

Now, divide both sides by \(-5\):

\[
x = -\frac{72}{5}
\]

So, the solution is:

\[
x = -\frac{72}{5}
\]

Would you like further details or have any questions?

Here are 5 questions related to this equation:
1. How do you combine fractions when solving algebraic equations?
2. What is the importance of eliminating fractions early in solving equations?
3. How does multiplying both sides of an equation help in simplifying it?
4. What strategies are helpful when dealing with negative coefficients?
5. Can you generalize this process for systems of equations?

**Tip:** When solving equations with fractions, multiplying through by the least common denominator (LCD) can make the process much simpler.

Learn how to solve the linear equation x - 3(3/2x + 3) = 3(-2/3x + 1) - 2/3x with detailed step-by-step instructions. This problem involves key algebraic concepts, including distributing, combining like terms, and working with fractions. Ideal for students in Grades 9-11.

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