Use the graphical method to find the value of x for which the expressions `(3x+2)/2` and `3/4x-2` are equal.

Answer

To find the value of \( x \) for which the expressions \( \frac{3x + 2}{2} \) and \( \frac{3}{4}x - 2 \) are equal using the graphical method, we can follow these steps:

### Step 1: Set the equations equal to each other
We start by setting the two expressions equal to each other:
\[ \frac{3x + 2}{2} = \frac{3}{4}x - 2 \]

### Step 2: Rewrite the equations in terms of \( y \)
Let’s define \( y \) for both expressions:
1. \( y = \frac{3x + 2}{2} \)
2. \( y = \frac{3}{4}x - 2 \)

### Step 3: Find points for the first equation
To plot the first equation, we will calculate \( y \) for a few values of \( x \):

- For \( x = 0 \):
\[ y = \frac{3(0) + 2}{2} = \frac{2}{2} = 1 \quad \text{(Point: (0, 1))} \]

- For \( x = 2 \):
\[ y = \frac{3(2) + 2}{2} = \frac{6 + 2}{2} = \frac{8}{2} = 4 \quad \text{(Point: (2, 4))} \]

- For \( x = -2 \):
\[ y = \frac{3(-2) + 2}{2} = \frac{-6 + 2}{2} = \frac{-4}{2} = -2 \quad \text{(Point: (-2, -2))} \]

### Step 4: Find points for the second equation
Now we will calculate \( y \) for the second equation:

- For \( x = 0 \):
\[ y = \frac{3}{4}(0) - 2 = -2 \quad \text{(Point: (0, -2))} \]

- For \( x = 4 \):
\[ y = \frac{3}{4}(4) - 2 = 3 - 2 = 1 \quad \text{(Point: (4, 1))} \]

- For \( x = 8 \):
\[ y = \frac{3}{4}(8) - 2 = 6 - 2 = 4 \quad \text{(Point: (8, 4))} \]

### Step 5: Plot the points on a graph
Now we will plot the points obtained from both equations on a graph:

- For the first equation, the points are:
- (0, 1)
- (2, 4)
- (-2, -2)

- For the second equation, the points are:
- (0, -2)
- (4, 1)
- (8, 4)

### Step 6: Draw the lines
Draw a line through the points of the first equation and another line through the points of the second equation.

### Step 7: Find the intersection point
Observe where the two lines intersect. This intersection point gives us the value of \( x \) for which the two expressions are equal.

### Final Result
From the graph, we find that the lines intersect at approximately \( x = -4 \).

Thus, the value of \( x \) for which the expressions \( \frac{3x + 2}{2} \) and \( \frac{3}{4}x - 2 \) are equal is:
\[ \boxed{-4} \]