Consider the cases when the inequality symbols point in the same direction and in opposite directions.
No
Practice makes perfect
For the solution set of a system of linear inequalities to be all real numbers, the intersection of the inequalities has to cover the entire coordinate plane. Given this piece of information, we know that none of the inequalities can have the symbols < or gt, as this would exclude numbers that are on the boundary line. Let's explore what happens if we have a system of inequalities where the inequality symbols are:
Pointing the same direction
Pointing opposite directions
Pointing the Same Direction
Consider the system of inequalities.
x≥ 0 x≥ 2
Let's graph both of these inequalities and shade accordingly.
As we are only looking for the intersection of the inequalities, we have to cut away the shading where both inequalities do not apply.
As we can see, x<2 is not a part of the solution set. No matter how we draw the inequalities, when the inequality symbols point in the same direction, we will never be able to cover the entire coordinate plane because we will always an unshaded region either to the right/left or above/below the inequalities where they do not overlap.
Pointing in Opposite Directions
Consider the system of inequalities
x≥ 0 x≤ 2
Let's graph both of these inequalities and shade accordingly.
Again, as we are only looking for the intersection of the inequalities, we have to cut away the shading where both inequalities do not apply.
As we can see here, x<0 and x>4 are not part of the solution set. No matter how we draw the inequalities, when the inequality symbols point in opposite directiond, we will never be able to cover the entire coordinate plane.
