Unit 4.3 The Solving Process

• Unit 4.2: Major focus on determining the number of solutions to linear and quadratic equations based on information about the coefficients of the variables in those equations

• Unit 4.3: Focus is on process and, unless otherwise stated, we seek only real solutions.

The scope of the discussion is expanded to include a wide variety of equations.

Equations in a single unknown x

• They have the from f(x) = g(x).• From this we can solve the equation

graphically (if the functions are reasonably well behaved) or numerically from a table of values.

• *These processes do not tend to give exact values (solutions) but are quite powerful because of their wide scope.

A quote

• “In fact, there is a sense in which school mathematics misleads students into believing that there exist methods which give exact solutions for all equations because these equations for which such methods exist are the only ones discussed.”

A better message

• It is more accurate to say that there exist general methods to provide approximate solutions to a wide variety of equations and special methods that provide exact solutions to a smaller, but a still large and important set of equations: linear, quadratic, etc.

Generalized Addition and Multiplication Properties of Equality

• It is important to identify the specific properties that enable equations to be solved in a systematic manner.

• In this section we examine in detail how these properties are used.