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:
[tex]y=mx+b[/tex]
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. [tex]y = 5x-1[/tex], [tex]y = 3x+4[/tex]
B. [tex]y = 5x-1[/tex], [tex]y = -3x+4[/tex]
C. [tex]y = 5x+1[/tex], [tex]y = -3x+4[/tex]
D. [tex]y = 5x-1[/tex], [tex]y = -3x-4[/tex]
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) [tex]y\ \textless \ 5x-1[/tex]
(2) [tex]3x+y \geq 4[/tex]
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) [tex]0\ \textless \ 9[/tex]
(2) [tex]6 \geq 4[/tex]
In fact, this is true.
