A44 APPENDIX Review A.6 Solving Equations Now Work the ‘Are You Prepared?’ problems on page A55. • Factoring Polynomials (Section A.3, pp. A27–A28) • Zero-Product Property (Section A.1, p. A4) • Square Roots (Section A.1, pp. A9–A10) • Absolute Value (Section A.1, p. A5) • Rational Expressions (Section A.5, pp. A35–A42) PREPARING FOR THIS SECTION Before getting started, review the following: OBJECTIVES 1 Solve Linear Equations (p. A45) 2 Solve Rational Equations (p. A47) 3 Solve Quadratic Equations by Factoring (p. A48) 4 Solve Quadratic Equations Using the Square Root Method (p. A49) 5 Solve Quadratic Equations by Completing the Square (p. A50) 6 Solve Quadratic Equations Using the Quadratic Formula (p. A51) 7 Solve Equations Quadratic in Form (p. A53) 8 Solve Absolute Value Equations (p. A54) 9 Solve Equations by Factoring (p. A54) An equation in one variable is a statement in which two expressions, at least one containing the variable, are equal.The expressions are called the sides of the equation. Since an equation is a statement, it may be true or false, depending on the value of the variable. Unless otherwise restricted, the admissible values of the variable are those in the domain of the variable. These admissible values of the variable, if any, that result in a true statement are called solutions , or roots , of the equation. To solve an equation means to find all the solutions of the equation. For example, the following are all equations in one variable, x: + = + = − − + = + = x x x x x x x 5 9 5 2 2 4 1 0 9 5 2 2 2 The first of these statements, + = x 5 9, is true when = x 4 and false for any other choice of x. That is, 4 is the solution of the equation + = x 5 9. We also say that 4 satisfies the equation + = x 5 9, because, when 4 is substituted for x, a true statement results. Sometimes an equation will have more than one solution. For example, the equation − + = x x 4 1 0 2 has −2 and 2 as solutions. Usually, we write the solutions of an equation as a set, called the solution set of the equation. For example, the solution set of the equation − = x 9 0 2 is { } −3, 3 . Some equations have no real solution. For example, + = x 9 5 2 has no real solution, because there is no real number whose square, when added to 9, equals 5. An equation that is satisfied for every value of the variable for which both sides are defined is called an identity . For example, the equation + = + + + x x x 3 5 3 2 2 is an identity, because this statement is true for any real number x. Solving Equations Algebraically One method for solving an equation is to replace the original equation by a succession of equivalent equations , equations having the same solution set, until an equation with an obvious solution is obtained.
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