What system of inequalities is represented by the graph? a. y < or equal to 5x -1 and -3x + y > or equal to 4 b. y < or equal to 5x -1 and 3x + y > or equal to 4 c. y < or equal to 5x + 1 and 3x + y > or equal to 4 d. y < or equal to 5x - 1 and 3x + y > or equal to -4

As shown in the figure, we have two straight line. One of them has a negative slope and the other has a positive one. In two dimensions, the equation for non-vertical lines is often given in the slope-intercept form by:



being m the slope of the line and b the y-intercept of it.

On the other hand, if x = 0 then y = b.

First of all we will order the equations above without inequalities like this:

A. ,
B. ,
C. ,
D. ,

 As shown in the figure b = -1 for one straight and b = 4 for the second one. This values take place when x = 0. So, we discard C and D, because if x = 0, then:

For C, b = 1 and b = 4
For D, b = -1 and b = -4

Let's analyze A and B. So:

For A, m = 5 and m = 3
For B, m = 5 and m = -3

Therefore, we discard A because of the statement above.

Finally the answer is B. So, the inequalities are:  

(1)
(2)

Let's prove this answer. We will take the point (2, 0) that is in the region in gray. So, substituting this point in the inequalities, we have:

(1) 
(2) 

In fact, this is true.

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