graphing inequalities?

Question

5x-3y is less than or = to 4
x+y<8
y>3

Answer 1

1) 5x - 3y <= 4

To solve this inequality, your first step would be to graph the line of the corresponding equality, 5x - 3y = 4, as a solid line.

5x - 3y = 4 implies
-3y = -5x + 4, so
y = (5/3)x - (4/3), which is in slope-intercept form and thus easy to graph. Once you've graphed this, you're going to either be shading above or below the line.

To determine where to shade, test the point (0,0); this means to make x = 0 and y = 0, plug this into your inequality, and see if you get a true or false statement.
If you get a true statement, shade the part of the line that includes (0,0). If false, shade the part of the line that does NOT include (0,0).

Test x = 0 and y = 0 for 5x - 3y <= 4. Then we get
5(0) - 3(0) <= 4, or 0 <= 4, which is a true statement. Shade the appropriate side of the line (which would be above, since it contains (0,0).

2) x + y < 8
Note that this time, our inequality is strictly less than, as opposed to less than _or equal to_. This means our graphed line will be dashed.

Refer to the corresponding equality, x + y = 8. This implies
y = -x + 8, so this is what you graph as a dashed line.

Test x = 0, y = 0, for x + y < 8. This would imply 0 < 8, which is a true statement, so again, shade the part of the line that contains (0,0). Graphically, you will see that it is below the line.

3) y > 3

Corresponding equality: y = 3.
y = 3 is a horizontal line (which you graph as "dashed"), and testing (0,0) for y > 3 would imply 0 > 3, which is FALSE. Shade above the dashed line y = 3.

Answer 2

You didn't say whether these were SEPARATE or JOINT conditions. However, the approach(es) to the first alternative (separate conditions) will help with the second case (joint conditions).

1. For "5x-3y is less than or = to 4", draw a SOLID line, y = (5/3) x - 4/3. All points ABOVE and to the LEFT of this line, INCLUDING points on the line itself, belong to this set. Shade this area in a distinctive way.

2. For "x+y < 8", draw a DASHED line y = 8 - x. All points BELOW and to the LEFT of this line , but NOT the points on the line, belong to this set. (A dashed line indicates non-inclusion.) Shade this area in a different, distinctive way.

3. For y > 3, draw a HORIZONTAL, DASHED line y = 3. All points ABOVE but NOT ON this line belong to this set. Shade this area in a different, distinctive way.

4, If you want only the points satisfying ALL THREE of these conditions, just identify the area containing all three distinctive shadings.

Live long and prosper.