`3y -5x lt 30`

Explanation

To solve the inequality \(3y - 5x < 30\), we will follow these steps:

### Step 1: Rearrange the inequality
First, we will rearrange the inequality to express \(y\) in terms of \(x\).

\[ 3y < 5x + 30 \]

Now, divide every term by 3 to isolate \(y\):

\[ y < \frac{5}{3}x + 10 \]

### Step 2: Identify the boundary line
The boundary line of the inequality is given by the equation:

\[ y = \frac{5}{3}x + 10 \]

### Step 3: Determine points for graphing
To graph the line, we can find two points.

1. **When \(x = 0\)**:
\[ y = \frac{5}{3}(0) + 10 = 10 \]
So, the point is \((0, 10)\).

2. **When \(y = 0\)**:
Set \(y = 0\) in the equation:
\[ 0 = \frac{5}{3}x + 10 \]
Rearranging gives:
\[ \frac{5}{3}x = -10 \quad \Rightarrow \quad x = -6 \]
So, the point is \((-6, 0)\).

### Step 4: Plot the points and draw the line
Now, we plot the points \((0, 10)\) and \((-6, 0)\) on the graph.

### Step 5: Draw the boundary line
Since the inequality is strict (\(<\)), we draw a dashed line through the points \((0, 10)\) and \((-6, 0)\). This indicates that points on the line are not included in the solution.

### Step 6: Shade the appropriate region
To determine which side of the line to shade, we can test a point not on the line. A good point to test is the origin \((0, 0)\):

\[ 3(0) - 5(0) < 30 \quad \Rightarrow \quad 0 < 30 \quad \text{(True)} \]

Since the inequality holds true, we shade the region that includes the origin.

### Final Result
The solution to the inequality \(3y - 5x < 30\) is the shaded region below the dashed line \(y = \frac{5}{3}x + 10\).

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