1. Graph the linear inequality.
y≤4x
2. Graph the linear inequality.
4x+5y≤20
To graph linear inequalities, follow these steps:
1. Start by graphing the corresponding equation. Treat the inequality sign as an equal sign.
For example, let's look at the first inequality: y ≤ 4x.
To graph the equation y = 4x, we need to find at least two points that satisfy it.
- Pick a value for x, let's say x = 0.
- Substitute this value into the equation to solve for y: y = 4(0) = 0.
- Plot the point (0, 0) on the coordinate plane.
- Repeat the process with another value of x, let's say x = 1.
- Substitute x = 1 into the equation: y = 4(1) = 4.
- Plot the point (1, 4) on the graph.
With these two points, you can draw a straight line passing through them.
2. Determine the region to shade based on the inequality sign:
- If the inequality is ≤ or ≥, use a solid line to indicate that the line itself is included in the solution.
- If the inequality is < or >, use a dashed line to indicate that the line itself is not included in the solution.
For the first inequality, y ≤ 4x, we use a solid line because it includes the line itself.
3. Determine which side of the line to shade by testing a point not on the line:
- Choose any point not on the line and substitute its coordinates into the inequality.
- If the inequality is true, shade the region containing that point; otherwise, shade the opposite region.
Let's use the origin (0, 0) as our test point and substitute it into the inequality y ≤ 4x:
0 ≤ 4(0)
0 ≤ 0
Since 0 is less than or equal to 0, the inequality is true, and we shade the region below the line.
Now onto the second inequality: 4x + 5y ≤ 20.
Follow the same steps as before:
- Graph the line corresponding to the equation 4x + 5y = 20.
- Find at least two points that satisfy it.
- Plot those points, draw the line passing through them, and use a solid line since the inequality is ≤.
Now, choose a point not on the line, such as (0, 0), and substitute it into the inequality:
4(0) + 5(0) ≤ 20
0 ≤ 20
Since 0 is less than or equal to 20, the inequality is true, and we shade the region below the line.
Remember to label your axes and indicate the shaded region correctly to complete the graph of each linear inequality.