Question

how to graph the inequality 2x-7y<-14

Answer 1

To graph the inequality \( 2x - 7y < -14 \), follow these steps:

1. **Rewrite the inequality**: First, it's often helpful to express the inequality in slope-intercept form (i.e., \( y = mx + b \)). Start by isolating \( y \):
\[
2x - 7y < -14
\]
Subtract \( 2x \) from both sides:
\[
-7y < -2x - 14
\]
Now, divide every term by \(-7\). Remember that dividing or multiplying by a negative number reverses the inequality sign:
\[
y > \frac{2}{7}x + 2
\]

2. **Graph the boundary line**: The boundary line is given by the equation \( y = \frac{2}{7}x + 2 \). To graph this line, you can find two points:

- **Y-intercept**: Set \( x = 0 \):
\[
y = \frac{2}{7}(0) + 2 = 2
\]
This gives you the point \( (0, 2) \).

- **X-intercept**: Set \( y = 0 \):
\[
0 = \frac{2}{7}x + 2 \implies \frac{2}{7}x = -2 \implies x = -7
\]
This gives you the point \( (-7, 0) \).

Plot these points and draw a line through them. Since the inequality is \( y > \frac{2}{7}x + 2 \) and not \( y \geq \frac{2}{7}x + 2 \), the line should be dashed to indicate that points on the line are not included in the solution.

3. **Test a point**: To determine which side of the line to shade, choose a test point not on the line. A common choice is the origin \( (0, 0) \). Substitute \( (0, 0) \) into the original inequality:
\[
2(0) - 7(0) < -14 \implies 0 < -14
\]
This statement is false, so we do not shade the side of the line that includes the origin.

4. **Shade the correct area**: Since the inequality is \( y > \frac{2}{7}x + 2 \), you would shade the area above the dashed line.

Your final graph should show a dashed line representing \( y = \frac{2}{7}x + 2 \) with the area above the line shaded.