Question

Graph the inequality on the axes below.
3, x, minus, y, is less than or equal to, 7
3x−y≤7

Answer 1

To graph the inequality \(3x - y \leq 7\), follow these steps:

1. **Convert the inequality to an equation**: Start by rewriting the inequality as an equation:
\[
3x - y = 7
\]

2. **Find the intercepts**: Find the x-intercept and y-intercept to help plot the line.
- **x-intercept**: Set \(y = 0\).
\[
3x - 0 = 7 \implies x = \frac{7}{3} \approx 2.33
\]
- **y-intercept**: Set \(x = 0\).
\[
3(0) - y = 7 \implies -y = 7 \implies y = -7
\]

The intercepts are \((\frac{7}{3}, 0)\) and \((0, -7)\).

3. **Plot the line**: Draw the line through the points \((\frac{7}{3}, 0)\) and \((0, -7)\). Since the inequality is less than or equal to (\(\leq\)), make the line solid.

4. **Shade the appropriate region**: Determine which side of the line to shade. Since the inequality is \(3x - y \leq 7\), you want to shade the region where the values of \(3x - y\) are less than or equal to 7. To test a point, use the origin \((0, 0)\):
\[
3(0) - 0 \leq 7 \implies 0 \leq 7
\]
The origin satisfies the inequality, so shade the region that includes the origin.

5. **Final graph**: On the graph, plot the solid line through the intercepts and shade the region below and including the line.

Here's a summary of the graphing procedure:
- Solid line from \((\frac{7}{3}, 0)\) to \((0, -7)\).
- Shade the area below this line (including the line itself).

If you need a visual representation, it is recommended to use graphing software or tools to create the actual graph.