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

Answer

To solve the problem using the graphical method, we need to find the value of \( x \) for which the expressions \( \frac{3x + 2}{2} \) and \( \frac{3x}{4} - 2 \) are equal. Here’s a step-by-step solution:

### Step 1: Define the equations
We start by defining the two expressions as equations:
1. \( y_1 = \frac{3x + 2}{2} \)
2. \( y_2 = \frac{3x}{4} - 2 \)

### Step 2: Create a table of values for \( y_1 \)
We will calculate several values of \( y_1 \) for different values of \( x \):

- For \( x = -2 \):
\[ y_1 = \frac{3(-2) + 2}{2} = \frac{-6 + 2}{2} = \frac{-4}{2} = -2 \]

- For \( x = 2 \):
\[ y_1 = \frac{3(2) + 2}{2} = \frac{6 + 2}{2} = \frac{8}{2} = 4 \]

- For \( x = 4 \):
\[ y_1 = \frac{3(4) + 2}{2} = \frac{12 + 2}{2} = \frac{14}{2} = 7 \]

### Step 3: Create a table of values for \( y_2 \)
Now, we will calculate several values of \( y_2 \) for the same values of \( x \):

- For \( x = -4 \):
\[ y_2 = \frac{3(-4)}{4} - 2 = -3 - 2 = -5 \]

- For \( x = 4 \):
\[ y_2 = \frac{3(4)}{4} - 2 = 3 - 2 = 1 \]

- For \( x = 8 \):
\[ y_2 = \frac{3(8)}{4} - 2 = 6 - 2 = 4 \]

### Step 4: Plot the points on a graph
Next, we plot the points for both equations on a graph:
- For \( y_1 \):
- Point 1: \( (-2, -2) \)
- Point 2: \( (2, 4) \)
- Point 3: \( (4, 7) \)

- For \( y_2 \):
- Point 1: \( (-4, -5) \)
- Point 2: \( (4, 1) \)
- Point 3: \( (8, 4) \)

### Step 5: Draw the lines
Draw straight lines through the points for both \( y_1 \) and \( y_2 \). The intersection point of these two lines will give us the value of \( x \) we are looking for.

### Step 6: Find the intersection point
From the graph, we can observe where the two lines intersect. This intersection point corresponds to the value of \( x \) where \( y_1 = y_2 \).

### Step 7: Solve algebraically (if needed)
To confirm the intersection point, we can set the two equations equal to each other and solve for \( x \):
\[ \frac{3x + 2}{2} = \frac{3x}{4} - 2 \]
Multiplying through by 4 to eliminate the denominators:
\[ 4 \cdot \frac{3x + 2}{2} = 3x - 8 \]
This simplifies to:
\[ 6x + 4 = 3x - 8 \]
Rearranging gives:
\[ 6x - 3x = -8 - 4 \]
\[ 3x = -12 \]
\[ x = -4 \]

### Final Answer
Thus, the value of \( x \) for which the two expressions are equal is \( x = -4 \).

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