System of Two Quadratic Inequalities (Classic)

This activity introduces the concept of the solution set of two simultaneous quadratic inequalities. That is, the set of all points that satisfy both inequalities simultaneously.

- The activity starts out with two equalities (equations). Explore the
possible solutions for this type of system.
- Select the "show intersection points" option to display the points where the two equations intersect.

- Experiment with the two equations by selecting the radio button next to
the one you want to change. Then use the sliders to change the coefficients.
Watch the intersection points.
- Can you find a pair of equations that yields more than two intersection points?
- Can you find a pair that yields only one intersection point?
- Can you find a pair that yields none?
- Can you do all three of the above using linear equations (a=0)?

- Select the first equality by clicking its radio button, and set its
coefficients (by typing or using the sliders) so that it reads, 'y = x - 2'.
- Change the inequality symbol to 'greater than'. All points in the area shaded green satisfy the inequality. Try identifying a point inside the shaded region and checking to see if its x and y coordinates satisfy the inequality.
- Try to predict what region will be shaded if you change the symbol to 'less than'. Try it. Were you right?
- Now use the 'greater than or equal to' and 'less than or equal to' symbols. How and why are these graphs different from those displayed when a strict inequality ( < or > ) is selected?
- How would you describe in general the shaded region for a linear inequality in slope-intercept form?
- Change the values of b and c to test your description. Does it hold true?

- Set first inequality back to equality and try working with the second
inequality. Type or use the sliders to create the equality (equation)'y =
x
^{2}+ 2' and then experiment with the different inequality options.- Does your general description of the shaded region for linear inequalities hold true for this quadratic? If not, what breaks down?
- Change the values of a, b, and c to explore the different possibilities. Come up with a description of the shaded region that works for both linear and quadratic inequalities.
- Can you mentally predict the shaded regions before you make your changes?

- Set the two inequalities back to 'y = x - 2' and 'y = x
^{2}+ 2'. Set the symbol for the first inequality to 'less than' and set the symbol for the second to 'greater than'.- What do you think the solution set to this system is?
- Enable the 'Highlight solution set' option to check to see if you were right. Were you?
- Disable 'Highlight solution set' and change the sign on one of the
equations, and try to identify the solution set again. Enable the highlight
again to check if you were right.
**Note:**The boundary lines for equations are always shown, but are not always in the solution set. How can you tell if the boundary line is part of the solution set? - Experiment with the different inequalities to see the different solution sets you can find. Can you predict the solution sets?

- Experiment with the two inequalities to see what solution sets you can
create.
- Can you make a solution set that is the empty set with two parabolas?
- Can you make a solution set that is completely enclosed by lines or curves?
- Can you make a solution set that is broken into two pieces?
- What role do the intersection points play in describing the solution set?