Systems of Inequalities

On 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

1. 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?
2. Change the line and see if your observations still hold. If not, refine them.
3. 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?
4. 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

1. 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.)
2. 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?
3. 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?
4. Describe the region, if any, in which A + Bx < y < C + Dx.
5. What happens when point P is on one of the lines? When P is at the intersection of the two lines?
6. 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 non-linear inequality. You can set the equations y = Ax² + 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, Ax² + Bx + C, and D + Ex computed for P.

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Investigate

1. 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.)
2. Drag point P around and see how its values for y, Ax² + Bx + C, and D + Ex compare to each other. What do you notice when you move point P to a different region?
3. Can you find a way to determine easily where y > Ax² + Bx + C, where y < Ax² + Bx + C, where y > D + Ex, and where y < D + Ex?
4. Describe the region, if any, in which Ax² + Bx + C > y > D + Ex.
5. What happens when point P is on one of the graphs? When P is at an intersection of the two graphs?
6. How is this sketch similar to the first two? How is it different?