Choose the correct graph of the following system of inequalities:

Question

Choose the correct graph of the following system of inequalities:

4x - 3y < -12
x - 6y >(line underneath) 6

 

Choose one answer.

 

A.

coordinate plane with a dashed line through points (0, -1) and (-6, -2). A solid line passes through points (0, 4) and (-3, 0). There is blue shading below the dashed line and yellow shading above the solid line. The shading turns green in the overlap.

 

B.

coordinate plane with a dashed line through points (0, -1) and (-6, -2). A solid line passes through points (0, 4) and (-3, 0). There is blue shading below the dashed line and yellow shading below the solid line. The shading turns green in the overlap.

 

C.

coordinate plane with a dashed line through points (0, 4) and (-3, 0). A solid line passes through points (0, -1) and (-6, -2). There is yellow shading above the dashed line and blue shading below the solid line. The shading turns green in the overlap

 

D.

coordinate plane with a dashed line through points (0, 4) and (-3, 0). A solid line passes through points (0, -1) and (-6, -2). There is yellow shading below the dashed line and blue shading below the solid line. The shading turns green in the overlap

Answer 1

Hi May,

 

Let me see if I can help you out. The first step would be to change the form the inequalities are in so you can see what kind of line they mean better. So you'll start with the first equation and do some basic algebra to move stuff around.

 

4x - 3y < - 12

-4x          -4x

-3y < -4x - 12

 -3         -3

y > 4x/3 + 4

 

Since the answers give you points where either the x- or y-coordinate are 0, we know these to be the intercepts of the line. From here we can then just write the inequality as an equation and plug in 0 for x to solve for the y-intercept. Then we can plug in 0 for y to solve for the x-intercept.

 

y = 4(0)/3 + 4                     0 = 4x/3 + 4

y = 0 + 4 = 4            (3/4)*-4 = 4x/3*(3/4)

 (0,4)                                   x= -3

   (-3,0)

 

The line that crosses these points is the line that is connected to the inequality y > 4x/3 + 4. Now, note that unlike the other inequality, this one does not have a line underneath the "greater than" sign. That means that there is no "equal" part to this inequality, and that is denoted on the graph by being drawn as a dashed line instead of a solid line. Thus you can already know that neither A nor B are the right answers, since those have a solid line through those points. 

 

Another thing to note is that our inequality has that y is "greater than," and so we know that y must be getting bigger. So we would shade in the upper area above the dashed line, since that is where the y is larger. The only answer that has both a dashed line going through (0,4) and (-3,0), as well as shading above the line, is answer C.

 

To make sure that is correct we would follow all these same steps for our second inequality. 

 

x - 6y ≥ 6

-x          -x

-6y ≥ -x + 6 

 -6         -6

y ≤ 1x/6 - 1

 

And then again, we solve for our x- and y-intercepts. 

 

y = 1(0)/6 - 1                     0 = 1x/6 - 1

y = 0 -1                             1 = 1x/6

y = -1                                6 = x

 (0,-1)                                 (6,0)    

 

Note that only one of the points we solved for was on our remaining line. To double-check that the remaining point on that line is a solution to our inequality, we would plug in (-6,-2).

 

-2 = 1(-6)/6 - 1

-2 = -1 - 1

-2 = -2

 

The points work and do not give me a false answer, for it is true that -2=-2, and so it is a solution to the inequality. So we know now that the line that passes through the points (0,-1) and (-6,-2) is the one that is given by our second inequality. Now, if you note, the second inequality is y ≤ 1x/6 - 1, so both "less than" and "equal to." This gets denoted in our graph by a solid line. It also has y being "less than" the rest of our inequality, and that is referring to y values that get smaller and smaller. So that would be below our line.

 

Finally, we put together everything we know. 

1. The line that passes through points (0,4) and (-3,0) must be dashed.
2. The area above the dashed line must be shaded.
3. The line that passes through points (0,-1) and (-6,-2) must be solid.
4. The area below the solid line must be shaded.

 

The only answer that continues to fulfill these requirements is answer C.