Question

Given the system y > -2 x + y < 4

Answer 1

Our system has two equations: y > -2 and x + y < 4.

A graph of y > -2 is a horizontal dashed line and everything above it is shaded. A graph of x + y < 4 shades everything to the left of it. From the graphs of this system you can determine an ordered pair that is true (so it is a solution) and an ordered pair that is false (not a solution).

A good test point - without a graph - is (0,0).

0 > -2 TRUE

0 + 0 = 0 and 0 < 4 TRUE

So (0, 0) is a solution in the system because it makes both equations true.

2. Now we want to find where the system is false. With a graph, it's the area unshaded by both. With the equations, we proceed as follows.

x + y < -4

y < - 4 - x Subtract x on both sides

Since y > -2 Then -2 < y. So by transitivity (if a < b and b < c, a < c), we have that

-2 < -4 - x

2 < -x add two to both sides

x > -2 multiply by -1 and change orientation.

So when x > -2 and y > -2 we have true inequalities - something we used above and found with (0,0). If we flip this around, we have false inequalities when x ≤ -2 or y ≤ -2. (The negation of an 'and' is an 'or'.)

So let's choose y to be -3.

-3 > -2 FALSE

x + y < 4

x + -3 < 4

x < 7

When x is < 7 it's true. To make it false, we need something bigger than 7. Let's use x = 10

10 + -3 < 4

7 < 4 FALSE

-3 > -2 FALSE

Thus, (10, -3) is not a solution because it makes both inequalities false.

Answer 2

you can come up with number to plug into the system for example numbers (7,10)

7 and 10 would give you 17 so is 17 greater than -2 yes

for #2 the answer would be no for(7,10)