Solving Linear Inequalities Graphically

A linear inequaility is alway of the form f(x) ≥ g(x). For example, in the equation 2x - 1 ≥ -2x + 5 we can regard f(x) as 2x - 1 and g(x) as -2x +5. Solving a linear inequality means transforming the original inequality into a new inequality that has the function x on one side of the inequality sign and a number (which is a constant function) on the other side. In this case the 'solution inequality' is x ≥ 1.5 (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. You may solve your inequality by dragging the RED, BLUE and BLACK dots on the graph in order to produce a 'solution equation' of the form x ≥ {constant function} or x ≤ {constant function} How do the expressions for f(x) and g(x) change as you drag the RED, BLUE and BLACK dots up and down? Can you explain the 'rule' that you must reverse the direction of the inequality when you multiply both sides by of the inequality by a negative numbr? Challenge - Dragging the BLACK dot changes both functions ,but dragging the RED dot changes only the RED function and dragging the BLUE dot changes only the BLUE function. This means that when you drag either the RED 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 the inequality? What is true about all the legitimate things you can do to a linear inequality?