To graph the inequality 2x + y < 4, follow these steps:
Step 1: Rewrite the Inequality
First, rewrite the inequality in slope-intercept form (y = mx + b). Start by isolating y:
y < -2x + 4
This tells us that the slope (m) is -2 and the y-intercept (b) is 4.
Step 2: Graph the Boundary Line
The boundary line of the inequality would be 2x + y = 4. First, we will graph this line:
- To find the y-intercept, set x = 0:
y = 4
2x = 4 → x = 2
The intercepts are (0, 4) and (2, 0). Plot these points on the graph.
Step 3: Draw the Boundary Line
Since the inequality is strict (<), draw a dashed line through the points (0, 4) and (2, 0). This indicates that points on the line are not included in the solution of the inequality.
Step 4: Shade the Appropriate Region
You now need to determine which side of the line to shade. Since the inequality is y < -2x + 4, you will shade below the line. This represents all the points (x, y) for which the inequality holds true.
Step 5: Verify with a Test Point
To confirm that you've shaded the right area, pick a test point that is not on the line, such as (0, 0):
2(0) + (0) < 4 → 0 < 4
This inequality is true, so the area containing (0, 0) should be shaded.
Final Graph
Your final graph should show a dashed line from (0, 4) to (2, 0), with the area beneath the line shaded to represent the solution to the inequality 2x + y < 4.
In conclusion, graphing the inequality involves plotting the boundary line, shading the correct region, and confirming your work with a test point. Happy graphing!