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THE RATIONAL NUMBERS 2-1 Rational Numbers A rational number is a terminating or repeating decimal. All rational numbers can be written as fractions. Multiplicative inverse: Converting a repeating decimal to a fraction: 2-2 Simplifying Rational Expressions A rational expression is a fraction where the numerator and denominator are polynomials. x 1 2 x2 4 x 4 3x Ex: 2 x x 1 2x y A rational expression is undefined when the denominator equals zero. 2 2 2 Ex: is undefined when x 1 because x 1 1 1 0 Like any fraction a rational expression can be simplified by canceling factors common to the numerator and the denominator. 12 3 4 3 Ex: 8 2 4 2 12 x 2 y 3 4 x x y 3x 8 xy 2 2 4 x y y 2y x2 x2 1 x 4x 4 x 2 x 2 x 2 2 ** Only terms that are being MULTIPLIED can be canceled. You CANNOT cancel terms that are being added or subtracted. x x x2 x 2 x cannot be simplified. x2 ** Always remember to factor completely before you cancel 2-3 Multiplying and Dividing Rational Expressions To multiply rational expressions, multiply the numerators and multiply the denominators. x 2 y y x 2 xy 2 y Ex: x y 3 x y 3 xy 3x Answers must be shown in simplest form. You can simplify after you multiply, but it is easier to cross-cancel before you multiply. 3x 6 y 2 3 y 3 x 2 y y 3 x 2 y 3 xy 2 y 3x 6 Ex: 2x 2x 3 x 2 y 3x 2 y To divide rational expressions, multiply the fist term by the reciprocal of the second term. x 3 x 3 x 3 4 2 2 2 Ex: 2x 4 2x x 3 2 x x 2-4 Adding and Subtracting Rational Expressions To add and subtract rational expressions you must first find common denominators. x 2 x 2 x x 2 x2 x Ex: x 2 x 2 x 2 x 2 x 2 x 2 x2 4 x 4 x2 2 x 2 x2 4 x 4 2 x 4x 4 x2 2x x2 4 2 x2 2 x 4 x2 4 2-5 Ratio and Proportion A ratio is a comparison of values. A proportion is an equation of two equal ratios. Cross-multiply to solve a proportion. x x 1 Ex: x2 x x x x 1 x 2 x 2 x 2 3x 2 0 3x 2 3 x 2 2 x 3 2-6 Complex Rational Expressions Complex fractions are fractions within fractions. Ex: 1 x x 1 2x 3 x1 2 3 7 x To simplify a complex fraction re-write the fraction as a division problem. Then divide and simplify. 1 1 1 1 1 x x 1 Ex: 2 x 1 x x x 1 x x Ex: 2x 3 x 1 2 2 x x 1 2 x 2 4x 3 2 3 x 1 3x 3 2-7 Solving Rational Equations There are several ways to solve rational equations. You can: 1. Work with fractions. 2. Multiply the equations by the LCD to remove fractions. 3. Combine fractions on each side of the equation to create proportion. **Any solution that causes fractions in the original equation to be undefined is called an extraneous solution. Extraneous solutions must be rejected. 2-8 Solving Rational Inequalities Rational inequalities can be solved using the same methods as rational equations. Remember that when you multiply or divide by a negative number the inequality symbol changes direction.