solve systems of linear inequalities?

Question

3x-y>6

x+y<6

Answer 1

You don't really "solve" the inequalities like you do with equations, because the solution set is much more simplified to describe. You can show the solution set though as a the shaded region of a graph. So graph each inequality and show where the solutions overlap.

Answer 2

Usually when you are asked to solve a system of two linear inequalities, they mean solve them graphically, drawing a graph to show where the solution region is.
To graph an inequality, you graph the line you would get if the inequality symbol were an equals sign, and then you shade one side. (Also, for a strict inequality, > or < instead of >= or <=, the line might be drawn as a dashed line or something like that.)
3x - y > 6:
add y to both sides, and subtract 6 from both sides
3x - 6 > y,
or
y < 3x - 6

x + y < 6:
subtract x from both sides, and you get
y < -x + 6

These lines are easy to graph, just plug in two points (such as the x- and y-intercepts) and draw the line through the points.
the line y = 3x - 6 goes through (0,-6) and (2,0).
the line y = -x + 6 goes through (0,6) and (6,0).
To figure out which side of each line to shade, try a point not on the line and see if it fits the equation.
So for x+y < 6, try (0,0) and see if it makes the inequality true. Do the same for the other inequality, and you can see where the separate inequalities are, and where they overlap, which is the solution.

Answer 3

Temporarily replace the inequality signs with equal signs and see what happens. Here
3x-y= 6 and x+y=6 can be added to get 4x=12 and x=3. Then y=3. Now you have to check each equation to find inequalties for each variable that apply to both equations at the same time. For example, if x=3, then y<3 works in equation 1 and also in equation 2. If you set y=3, your result for x would not be consistent.