Question

What system of inequalities is represented by the graph?

Answer 1

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:

$y=mx+b$

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. $y = 5x-1$, $y = 3x+4$
B. $y = 5x-1$, $y = -3x+4$
C. $y = 5x+1$, $y = -3x+4$
D. $y = 5x-1$, $y = -3x-4$

 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) $y\ \textless \ 5x-1$
(2) $3x+y \geq 4$

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) $0\ \textless \ 9$
(2) $6 \geq 4$

In fact, this is true.