Answer:
See attached graphs Graph1, Graph2 and Graph3 and detailed explanation below
Step-by-step explanation:
The process for plotting and shading are essentially the same but with different inequalities
To plot any line
- Plug in a value for x and find the corresponding y value
- Repeat for another x value and find the corresponding y value
- Draw a straight line through the two points
To determine area to shade
Choose a point on either side of the line from the graph. See if the corresponding (x,y) values satisfy the inequality. If they do, then that's the area to shade. If not, shade the other area
1. Inequality y > -3x - 2
The line equation is y = -3x -2
Let's choose x = 0. Then y = -3(0) -2 = -2. So one point is (0, -2)
Choose another value for x, say x = 1
y = -3(1) - 2 = -3-2 = -5. So another point is (1,-5).
Draw a straight line through both points (see attached Graph1)
Determining area to shade
Choose a point on the graph on one side of the line. Let's choose (1,2). So x = 1 and y = 2. Does this satisfy the inequality y = -3x -2? Let's see
Is 2 > -3(1) - 2 ?
2 > -3-2 ?
2 > -5 ? YES since a positive number is greater than a negative number. So the area containing (1,2) should be shaded as shown in Graph 1
2. Inequality 4x + 5y ≤ 20
Line equation is 4x + 5y =20
Convert to a function of x as follows
- Subtract 4x from both sides ==> 5y = 20 -4x
- Divide by 5 both sides ==> y = 20/5 - (4/5)x ==> y = 4- 4x/5
Choose two values of x and find corresponding y values to get the coordinates of two points
Plug x = 0 into line equation ==> y = 4- 4x/5
==> y = 4 - 4(0)/5 = 4
==> (0, 4) is one point
Choose another value of x, say 5(this is a convenient value since we are dividing by 5)
==> y = 4- 4(5)/5 = 4-4 = 0
==> (5, 0) is another point
Draw straight line through (0,4) and (5,0)
To determine area to shade let's choose a point on the graph. (0,0) is a convenient point. Do the values of x=0 and y=0 satisfy the inequality
4x + 5y ≤ 20?
Plugging in x = 0 and y = 0 gives us
4(0) + 5(0) = 0 and 0 is indeed ≤ 20
So the inequality is satisfied at (0,0)
Shade the region on the side of the line where (0,0) is (see Graph2)
3. Inequality y > -4
The line equation is y = -4
Since this equation does not involve x, the line plot will be a horizontal line at y = -4 for all x values
Simply draw a horizontal line which crosses the y axis at y = -4
To determine which area to shade, choose a point. Use (0,0)
Does y = 0 satisfy the inequality y > -4?
0 > -4 so indeed the point (0,0) satisfies the inequality. Just shade the area on the side of the line which contains (0,0). This will be the region above the line y = -4 See Graph3
Hope that helps :)
