System of inequalities
1. Type of line
We know that:
<, > represent a dotted line
≤, ≥ represent a solid line
Inequality 1: dotted line
Inequality 2: solid line
2. y- intercept of the limit line
We know that the equation of a line is given by
y = mx + b,
where b is a number: b is its y-intercept.
Inequality 1: we have that its limit line is
y = x - 2
Using the line equation we have that its y-intercept is -2.
Inequality 2: we have that its limit line is
y = -2x +3
Using the line equation we have that its y-intercept is 3.
Now we can each line y -intercept:
3. Slope of the limit line
We know that the equation of a line is given by
y = mx + b,
where m is a number: m is the slope of the line and it represents its growth.
Inequality 1: we have that its limit line is
y = x - 2
Using the line equation we have that its slope is 1.
This means that when we go 1 unit to the right it will go 1 unity up
Inequality 2: we have that its limit line is
y = -2x +3
Using the line equation we have that its slope is -2.
This means that when we go 1 unit to the right it will go 2 unities down.
4. Line
Now, that there are two dots, we will have the graph drawn:
5. Shaded region
In order to decide which side of the lines should be the shaded region we are going to select the origin point: (0, 0). We are going to evaluate if it is possible for each inequality that they will have an answer that crosses that point.
In order to find we just replace x=0 and y=0 and see if it has or not a logical answer:
Inequality 1:
y < x - 2
↓ replacing x=0 and y=0
0 < 0 - 2
↓ 0 - 2 = -2
0 < -2
It is not true that -2 is higher than 0, then this inequality doesn't have the origin point on its shaded region:
Inequality 2:
y ≥ -2x + 3
↓ replacing x=0 and y=0
0 ≥ -2 (0) + 3
↓ -2 (0) + 3 = 0 + 3 = 3
0 ≥ 3
It is not true that -3 is higher than 0, then this inequality doesn't have the origin point on its shaded region neither:
