Question

What is the answer ?

Answer 1

The system of inequalities is the following:

i) y ≤ –0.75x
ii)y ≤ 3x – 2

since $0.75= \frac{75}{100}= \frac{3}{4}$, we can write the system again as 

$i) y \leq - \frac{3}{4}x$
$ii) y \leq 3x-2$

Whenever we are asked to sketch the solution of a system of linear  inequalities, we:

1. Draw the lines
2. Color the regions of the inequalities.
3. The solution is the region colored twice.


A.

to draw the line $y =- \frac{3}{4}x$

consider the points: (-4, 1) and (0, 0), or any 2 other points (x,y) for which $y =- \frac{3}{4}x$ hold.

since we have an "smaller or equal to" inequality, the line is a solid line (not dashed, or dotted).

In order to find out which region of the line to color, consider a point not on the line, for example P(1, 1), which is clearly in the upper region of the line.

For (x, y)=(1, 1) the inequality $y \leq - \frac{3}{4}x$, does not hold because 

$1 \leq - \frac{3}{4}*1= -\frac{3}{4}$ is not true,

this means that the solution is the region of the line not containing (1, 1), as shown in picture 1.


B.
similarly, to draw the solution of inequality ii) y ≤ 3x – 2, 

we first draw the line y=3x-2, using the points (0, -2) and (2, 4), or any other 2 points (x,y) for which y=3x-2 holds.

after we draw the line, we can check the point P(1, 7) which clearly is above the line y=3x-2.

for (x, y) = (1, 7), the inequality y ≤ 3x – 2 does not hold

because 7 is not ≤ 3*1-2=1, so the region we color is the one not containing P(1, 7), as shown in picture 2.


The solution of the system is the region colored with both colors, the solid lines included. Check picture 3.

the lines intersect at (0.533, -0.4) because:

–0.75x=3x-2
-0.75x-3x=-2
-3.75x=-2, that is x= -2/(-3.75)=0.533

for x=0.533, y=3x-2=3(0.533)-2=-0.4

Answer: Picture 3, the half-lines included. So the graph is in the 3rd and 4th Quadrants