Which is the graph of the linear inequality 2x – 3y < 12?
The Correct Answer and Explanation is :
To graph the linear inequality (2x – 3y < 12), we first rewrite it in slope-intercept form, which is useful for graphing. Here’s how to do it step-by-step:
Step 1: Rewrite the Inequality
Start with the inequality:
[
2x – 3y < 12 ] To isolate (y), we can subtract (2x) from both sides: [ -3y < -2x + 12 ] Next, divide every term by (-3). Remember, when we divide by a negative number, we must reverse the inequality sign: [ y > \frac{2}{3}x – 4
]
Step 2: Graph the Boundary Line
The corresponding equation for the boundary line is:
[
y = \frac{2}{3}x – 4
]
To graph this line, we can identify two points.
- Y-intercept: Set (x = 0):
[
y = -4 \quad \Rightarrow \quad (0, -4)
] - X-intercept: Set (y = 0):
[
0 = \frac{2}{3}x – 4 \quad \Rightarrow \quad \frac{2}{3}x = 4 \quad \Rightarrow \quad x = 6 \quad \Rightarrow \quad (6, 0)
]
Plot these two points on a coordinate plane and draw a dashed line through them. The dashed line indicates that points on the line itself are not included in the solution (since the inequality is strict: (<)).
Step 3: Shade the Correct Region
Since the inequality is (y > \frac{2}{3}x – 4), you will shade the region above the line. This region represents all the points where the (y)-values are greater than those on the line.
Conclusion
The graph of the inequality (2x – 3y < 12) consists of a dashed line with slope (\frac{2}{3}) that passes through the points (0, -4) and (6, 0). The solution set is all the points above this line, indicating the values of (x) and (y) that satisfy the inequality.