Determine if the ordered pairs two comma negative two and negative three comma negative four
are solutions to the system of linear inequalities.
If an ordered pair is a solution it must satisfy both inequalities or make both inequalities
true.
It is not enough for the ordered pair to satisfy just one of the inequalities, it must satisfy
both inequalities.
So to see if the ordered pair two comma negative two is a solution, we'll substitute two for
x and negative two for y into both inequalities and see if both inequalities are true.
So we'll first perform the substitutions into the inequality x minus three y is greater
than or equal to four.
This would give us two minus three times negative two greater than or equal to four.
Now we need to simplify the left side.
Three times negative two is equal to negative six.
Minus negative six is the same as plus positive six.
So the left side simplifies to two plus six.
Two plus six is eight.
So the inequalities simplifies to eight greater than or equal to four, which is true.
Which means the ordered pair does satisfy the first inequality.
It does not mean the ordered pair is a solution to the system.
To be a solution to the system, it must also satisfy the second inequality.
So now we'll perform substitution into the inequality two x minus y less than one.
Performing the substitution we would have two times two minus negative two less than
one.
Two times two is four.
Minus negative two is the same as plus positive two.
So the left simplifies to four plus two which is six.
Six less than 1 is not true or is false.
So because the ordered pair does not satisfy both inequalities, the ordered pair is not
a solution to the system.
Now let's check the ordered pair negative three comma negative four.
Now we'll substitute negative three for x and negative four for y, again in both inequalities.
Performing the substitution in to the first inequality, we would have negative three minus
three times negative four greater than or equal to four.
Now were were are going to have minus negative twelve which is the same as plus twelve.
So this simplifies to negative three plus 12 greater than or equal to four.
Negative three plus twelve is equal to positive nine.
So the inequalities simplifies to nine greater than or equal to 4, which is true.
The ordered pair does satisfy the first inequality.
Let's see if it satisfies the second inequality.
Performing the substitution, we would have two times negative three minus negative four
less than 1.
Two time negative three is negative six.
So we have negative six minus negative four is the same as plus positive four.
So we have negative six plus four less than one.
Negative six plus four is negative two.
Negative two less than one is true.
So the ordered pair does satisfy both inequalities and therefore negative three comma negative
four is a solution to the system.
Let's verify these results graphically.
The solution to the system of inequalities is this double shaded region here.
Notice how this point is in the double shaded region.
This point has the ordered pair negative three comma negative four, which is a solution to
the system.
Notice this point is not in the double shaded region and therefore is not a solution.
This point is given by the ordered pair two comma negative two, which is not a solution
to the system.
I hope you found this helpful.