Linear Inequalities in Two Variables - University of Notre Dame
Linear Inequalities in Two Variables
The next topic we will study is optimization ¡ª how to
make the most of limited resources. This will lead us to
situations like the following: if apples cost $1.45 per kilo
and pears cost $1.25 a kilo, what combination of apples and
pears can I buy with at most $5?
If I buy a kilos of apples and p kilos of pears then I spend
1.45a on apples and 1.25p on pears, so 1.45a + 1.25p in
total. So whatever combination I buy, it must satisfy
1.45a + 1.25p ¡Ü 5.
This is an example of a linear inequality.
Review of lines
The equation of a line is given by:
ax + by = c.
for some given numbers a, b and c.
A vertical line which runs through the point c on the x-axis
has equation
x = c.
A horizontal line which runs through the point d on the
y-axis has equation
y = d.
A line which runs through the point (0, 0) has an equation
of the form
ax + by = 0.
Review of lines
Minor technical issue: if a = b = 0 then either
I all points satisfy the equation (if c = 0) or
I no points satisfy the equation (if c 6= 0).
Whenever we discuss the line ax + by = c we agree that not
both a and b are 0.
Given an equation of a line, its graph is the set of all points
in the xy-plane which satisfy the equation.
In particular the graph is an example of a set and we can
form unions, complements, intersections, etc.
From plane geometry you know that the intersection of two
lines is either the empty set (the lines are parallel), or the
line (the lines are equal) or a single point.
You can find this single point (if it exists) with a little
algebra.
Review of lines
Example: The line 2x + 3y = 6.
The point (0, 2) satisfies the equation 2x + 3y = 6, because
2(0) + 3(2) = 6. Hence the point (0, 2) is on the graph of
that equation, so is on the line.
The point (0, 0) does not satisfy the equation 2x + 3y = 6,
since 2(0) + 3(0) 6= 6. Hence the point (0, 0) is not on the
graph of the equation 2x + 3y = 6, and is not on the line.
Review of lines
We can draw the graph of a line if we know the location of
any 2 points on the line. The x- and y-intercepts (the
places where the line hits the x-axis and the y-axis) are
usually the easiest points to find.
I To find the x-intercept, we set y = 0 in the equation
and solve for x.
I To find the y-intercept, we set x = 0 in the equation
and solve for y.
Given these two points (or any two distrint points) we can
draw the line by joining the points with a straight edge and
extending.
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