Systems of InequalitiesOn this web page you will investigate the
systems of linear and nonlinear inequalities. This exploration will
reinforce the concepts in Lesson 6.5 of Discovering Advanced
Algebra: An Investigative Approach.
Sketch
This sketch shows the graph of the equation y
= A + Bx. Each side of the line is colored
differently. You can change the equation by moving the sliders for A
and B. As you adjust the line or point P, the
sketch will recalculate the values of y and A + Bx
(where x and y are the coordinates of P).
Investigate
 Move point P around and compare the value of y
to the value of A + Bx. What happens when P
is on the line? What happens when you move P across the line?
What happens when P is very close to the line?
 Change the line and see if your observations still hold. If not,
refine them.
 What can you say about the solutions of the linear inequality y
> A + Bx? What are the solutions of the linear
inequality y < A + Bx?
 Can you find a simple way to decide which region represents y
> A + Bx and which region represents y < A +
Bx?
Sketch
The sketch below will help you solve a system of two
linear inequalities. You can set the equations y = A + Bx
and y = C + Dx using the sliders for A, B,
C, and D. As you drag point P, you will
see the values of y, A + Bx, and C + Dx
computed for P.
Investigate
 When the two lines intersect, into how many regions do they
separate the plane? (Note that sometimes the intersection point will
lie off the screen.)
 Drag point P around and see how the values of y,
A + Bx, and C + Dy for P compare to
each other. What do you notice when you move point P
to a different region?
 Can you find a way to determine easily where y > A +
Bx, where y < A + Bx, where y >
C + Dx, and where y < C + Dx?
 Describe the region, if any, in which A + Bx < y
< C + Dx.
 What happens when point P is on one of the lines? When P
is at the intersection of the two lines?
 When the two lines are parallel, into how many regions do they
separate the plane? What can you say about A + Bx and C
+ Dx for point P in these regions?
Sketch
The sketch below will help you solve a system of one
linear inequality and one nonlinear inequality. You can set the
equations y = Ax^{2} + Bx + C and
y = D + Ex using the sliders for A, B,
C, D, and E. As you drag point P,
you will see the values of y, Ax^{2} +
Bx + C, and D + Ex computed for P.
Investigate
 When the two graphs intersect, into how many regions might they
separate the plane? (Note that sometimes you will not be able to see
the intersections, and sometimes the graphs will not intersect.)
 Drag point P around and see how its values for y,
Ax^{2} + Bx + C, and D
+ Ex compare to each other. What do you notice when you move
point P to a different region?
 Can you find a way to determine easily where y > Ax^{2}
+ Bx + C, where y < Ax^{2}
+ Bx + C, where y > D + Ex,
and where y < D + Ex?
 Describe the region, if any, in which Ax^{2}
+ Bx + C > y > D
+ Ex.
 What happens when point P is on one of the graphs?
When P is at an intersection of the two graphs?
 How is this sketch similar to the first two? How is it
different?
