# 1-4 - CHAPTER 1 BASIC ALGEBRA TECHNIQUES 1.4 SOLVING...

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CHAPTER 1: BASIC ALGEBRA TECHNIQUES § 1.4 SOLVING EQUATIONS AND INEQUALITIES Solving Polynomial Equations: An equation is a statement of equality between two expressions, and this may or may not be true for some values of the variable(s). Unless specified otherwise, we assume the variables are real [i.e., take on real number values only]. Thus x 2 y 2 1 is an equation for which no values of x or y make it true; x 2 0 has one solution; and x 2 y 2 1 has an infinite number. A particular choice of values of the variable(s) making an equation true is called a solution or root, and the collection of all solutions is called the solution set. In order to solve equations or inequalities [a statement of inequality between two expressions with solutions and solutions sets defined as for equations], we use a number of basic principles: 1. Addition Principle: Adding an expression which is defined everywhere to both sides yields an equivalent [i.e., has exactly the same solution set] equation or inequality. Wherever the added expression is undefined, the original, equation or inequality must be checked. 2. Multiplication Principle: Multiplying an equation by an expression that is defined everywhere and never equal to zero yields an equivalent equation. Wherever the new expression is 0 or undefined, you must check the original equation [if it is undefined – it possibly destroys roots; if it is 0 – it can create roots]. For inequalities, if the multiplying expression is 0 the inequality’s direction is preserved, and if it is 0 then the inequality is reversed. 3. Power Rule (a simple version): If all variables and expressions are positive then exp1 exp2 if and only if exp1 exp2 . We must be careful in case the variables and expressions can be negative. As further refinements are required, they will be introduced. 4. 0-products rule: ab 0 if and only if at least one of a , b is zero. Thus, for an equation in one variable, to solve we generally follow BEDMAS in reverse order, and as a value/term is moved across the equal sign, we perform the “opposite operation”. examples: 2 x 3 6 x 2 x x 6 3 3 x 9 3 x 3 9 3 x 3 x 2 2 x 8 0 x 4 x 2 0 x 4 0 or x 2 0 x 4 x 2 x 2 6 x 3 x 3 2 9 3 x 3 2 12 x 3 12 x 3 12 Note that in the third example, we don’t know if the or the root is required, and so we must keep both. Generally we can solve quadratics [polynomials of degree 2, in one variable] by the quadratic formula: ax 2
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