Linear Inequalities solved w/ graphs & symbols
- Judah L Schwartz
A linear inequality is always of the form . For example, in the inequality we can regard as and as Solving a linear inequality means transforming the original inequality into a new inequality that has the function on one side of the equal sign and a number (which is a constant function) on the other side. In this case the 'solution equation' is (why is 1.5 a function?) The app allows you to enter a linear function f(x) = mx + b by varying m and b sliders and a function g(x) = Mx + B by varying M and B sliders. The fundamental question this applet poses is FOR WHAT VALUES OF x IS THE GREEN FUNCTION LARGER THAN THE BLUE FUNCTION? You may solve your inequality graphically by dragging the GREEN, BLUE and WHITE dots on the graph in order to produce a 'solution inequality' of the form . Why does the applet behave as it does if m = M? Why does it behave as it does when m=M and b=B ? Challenge - Dragging the WHITE dot changes both functions, but dragging the GREEN dot changes only the GREEN function, and dragging the BLUE dot changes only the BLUE function. This means that when you drag either the GREEN dot or the BLUE dot you are changing only ONE side of the inequality!! - Why is this legitimate? - Why are we taught that you must do the same thing to both sides of an inequality? - What is true about all the legitimate things you can do to a linear inequality? - What are the symbolic operations that correspond to dragging each of the dots? You may also solve your inequality symbolically by using sliders to change the linear and constant terms on each side of the inequality. - What are the graphical operations that correspond to each of the sliders? Particular attention should be paid to the behavior of the SCALE slider. What other questions could/would you pose to your students based on this applet ?