A linear inequality is always of the form [math]f(x) >= g(x)[/math]. For example, in the inequality [math]2x - 1 >= -2x + 5[/math] we can regard [math]f(x)[/math] as [math]2x - 1[/math] and [math]g(x)[/math] as[math] -2x +5.[/math][br][br]Solving a linear inequality means transforming the original inequality into a new inequality that has the function [math]x[/math] on one side of the equal sign and a number (which is a constant function) on the other side. [br][br]In this case the 'solution equation' is [math]x >= 1.5[/math] (why is 1.5 a function?)[br][br]The app allows you to enter a linear function f(x) = mx + b by varying m and b sliders and [br]a function g(x) = Mx + B by varying M and B sliders.[br][br][b]The fundamental question this applet poses is [br][br]FOR WHAT VALUES OF x IS THE [i][color=#0a971e]GREEN[/color] [/i]FUNCTION LARGER THAN THE [i][color=#1551b5]BLUE[/color][/i] FUNCTION?[/b][br][br]You may solve your inequality [size=150][u][i][b]graphically[/b][/i][/u][/size] by dragging the [color=#00ff00][i][b]GREEN[/b][/i][/color], [color=#1e84cc][i][b]BLUE[/b][/i][/color] and [i][b]WHITE[/b][/i] dots on the graph in order to produce a 'solution inequality' of the form [math]x >= {constant function}[/math]. [br] Why does the applet behave as it does if m = M? [br] Why does it behave as it does when m=M and b=B ?[br][br][i][b]Challenge[/b][/i] - Dragging the [i][b]WHITE[/b][/i] dot changes both functions, but[br] dragging the [color=#00ff00][i][b]GREEN[/b][/i][/color] dot changes only the [color=#00ff00][i][b]GREEN[/b][/i][/color] function, and [br] dragging the [color=#1e84cc][i][b]BLUE[/b][/i][/color] dot changes only the [color=#1e84cc][i][b]BLUE[/b][/i][/color] function.[br][br]This means that when you drag either the GREEN dot or the BLUE dot you are changing only [b][i]ONE[/i][/b] side of the inequality!![br][br]- Why is this legitimate? [br][br]- Why are we taught that you must do the same thing to both sides of an inequality?[br][br]- What is true about all the legitimate things you can do to a linear inequality?[br][br]- What are the symbolic operations that correspond to dragging each of the dots?[br][br]You may also solve your inequality [size=150][u][i][b]symbolically[/b][/i][/u][/size] by using sliders to change the linear and constant terms on each side of the inequality.[br][br]- What are the graphical operations that correspond to each of the sliders?[br][br]Particular attention should be paid to the behavior of the SCALE slider.[br][br][i][b][color=#ff0000]What other questions could/would you pose to your students based on this applet ?[/color][/b][/i]