Problem 7 in Week 1 of Unit 3 Course 2
- Kenneth Wright
Think of equations or inequalities as statements that may or not be true, depending upon the value of the variable x. From this perspective, each side of the statement is a function and the value of x may be any value in the domain of x (usually all real numbers). Solutions are values of x for which the statement is true. There may be many values for which the statement is false. The statement is intelligible and meaningful in either case. By convention, we map all the positions of solutions on the number line to represent the set of solutions. This applet animates statements of the form: f(x) > g(x). Here f(x) is an absolute value function and g(x) is a constant, but the principles demonstrated here apply to any comparison of functions. To be specific, each function must have the same set as its domain, which here is the set of all real numbers. The real number line is shown here to be along the x-axis of the plane.
The approach here should not be confused with a common topic in algebra courses in which one locates points on the plane as solutions. You can view a related problem from this perspective by toggling the switch to the (x,y) solutions view. Note that if we look at things this way, we mark POINTS ON THE PLANE for which all inequalities are true. It is possible to describe relevant functions of x and y for which these points would be solutions, but we do not consider such functions in a formal way until Course 3.