Section 3-3: Polynomial
Inequalities in Two Variables
Demo: Linear Inequalities (Larry Green)
The graphs of inequalities
in two variables consist of points in the x-y graphing plane. Two
find the graph, first graph the equation.
This is the boundary line
between what makes the inequality greater than or less than. The
line can be solid (points on the line make
the statement true) or dotted (does
not include the points on the line).
You must decide which side of the line to shade. The easiest way
to tell is to pick a point on one side of the line. If it makes the
statement true, you have the correct side. Go ahead and shade it.
If it makes the statement false, you have the wrong side. Shade the
the inequality y < 2x + 1
The boundary is a line.
The y-intercept is (0,1), with slope of 2. Graph the line!
Notice the line is dotted
because the line is not included ( no or equal to in the statement).
Now pick a point not
on the line. Use (0,0). Is 0 < 2(0) + 1? Sure is,
so shade this side of the line!
2) Graph the solution
of y > x + 3 and y < 9 - x2
The first has a boundary
line. The second boundary is a parabola. The line graphs with
slope 1 and y-intercept (0, 3). The parabola has a vertex point at
(0, 9) and crosses the x-axis at 3 and -3. Look back to graphing
parabolas if you forgot about that!!
Now pick a point.
Try (0, 0). Try the line. 0 > 0 + 3. This is false.
Shade the other direction, above the line. Now try (0, 0) in the
0 < 9 - 0.
This is true. Shade inside the parabola. Viola!! You
The solution is the cross-hatched
3) Find the solution
for |y| > 1 and |x| < 3 in two variables.
Both are absolute value
graphs The first can be written as y > 1 or y <
-1. The second can be written as -3 < x < 3.
The first absolute value
graph means we want values above one and below -1 including the equality.
The second absolute value
means x must be between -3 and 3 including these points. Add that
The solution is the crossed-hatched
4) Find the solution
for y < x + 1, y > -x + 4 and y > 2
Each of these has a boundary
which is a line. Two are slanted and the other is horizontal.
The first has slope 1 and y-intercept 1. The second has slope -1
and y-intercept 4. The last is a horizontal line at 2.
You can try (0,0) for
all three lines. In the first line 0 < 0 + 1 is true.
Shade below the blue line. In the second line, 0 > 0 + 4 is
false. Shade above the green line. In the third line, 0 >
2 is false. Shade above the purple line. The result is:
On to linear programming:
Back it up dude!! I'm