1 Day : Section P-6 Rational Epressions; Section P-7 Equations Rational Epressions A rational epression (Fractions) is the quotient of two polynomials. The set of real numbers for which an algebraic epression is defined is the domain of the epression. Eamples of rational epressions: 5 6 (a) (b) 9 The domain of (a) is all real numbers ecept for =. In interval notation you would write,, The domain of (b) is all real numbers ecept for = and. In interval notation you would write,,, To find the domain set the denominator equal to zero and solve the resulting equation. Your answer to this equations is how your domain is restricted. Practice: In 1-, State the domain in interval notation. 7 1.. 5 15. 445 Simplifying Rational Epressions A rational epression is in simplified form provided its numerator and denominator have no common factors (other than ±1). To simplify a rational epression: 1. Factor the numerator and denominator completely.. Divide both the numerator and the denominator by any common factors. Practice: In 1-, Simplify. Also state the domain in interval notation. s 4s1 y y 1... s 4 y 1 1 1
Multiplying Rational Epressions To Multiply Rational Epressions use the following steps: 1. Factor all numerators and denominators completely.. Divide numerators and denominators by common factors (aka SIMPLIFY).. Multiply the remaining factors in the numerators and multiply the remaining factors in the denominators. Practice: In 1-, Multiply and simplify. 6 1. 4 6 9. 4 1 8 1. 5 y 6 y y 10y Dividing Rational Epressions The quotient of two rational epressions is found by inverting the divisor and multiplying. Practice: In 1-, Divide and simplify. 5 1. 1 6 8. y y y y 1 y y y Adding and Subtracting Rational Epressions with Different Denominators. In order to add or subtract rational epressions they must have common denominators. So we need to be able to find the Least Common Denominator. Finding the Least Common Denominator 1. Factor each denominator completely.. List the factors of the first denominator.. Add to the list in step any factors of the second denominator that do no appear in the list. 4. Form the product of each different factor from the list in step. This product is the least common denominator. Practice: In 1-, Find the least common denominator. 1. and 7 6 9 9 5 u 6u 4u 1u. and
Adding and Subtracting Rational Epressions That Have Different Denominators 1. Find the LCD of the rational epressions.. Rewrite each rational epression as an equivalent epression whose denominator is the LCD. To do so, multiply the numerator and the denominator of each rational epression by any factor(s) needed to convert the denominator into the LCD.. Add or subtract numerators, placing the resulting epression over the LCD. 4. If possible, simplify the resulting rational epression. Practice: In 1-, Perform the indicated operation. 6 4 4 1.. 4 5 6 9. 4 6 5 1 Simplifying a Comple Rational Epression. A comple rational epression is a faction that contains a fraction in its numerator or denominator. To simplify a comple fraction, write its numerator and its denominator as single fractions. Then divide by multiplying the reciprocal of the denominator. Practice: In 1-, Simplify each comple rational epression. 1 1 1 1.. 1. 4 7 4 Equations Solving an equation in involves determining all values of that result in a true statement when substituted into the equation. Such values are solutions, or roots, or the equation. Practice: Solve and check: 1. 4(+1) = 9 + ( 5)
4 Solving Rational Equations In order to solve rational equations use the following steps: 1. Identify values of that make denominators zero. (State the domain). Find the LCD of all denominators in the rational equation.. Multiply both sides of the rational equation by the LCD. 4. Simplify. The equation is cleared of fractions. 5. Solve the new equation found in step 4. Practice: In 1-, Solve. 1 11 1. 5 1 1. 1 5 6 4 8 Solving a Formula for a Variable When solving a formula for a variable or goal is to isolate the variable that is asked to be solved for. Practice: Solve for q: 1 1 1 p q f Solving Equations Involving Absolute Value To solve absolute value equations: 1. Isolate the absolute value epression. Drop the absolute value notation and write two equations: one positive, one negative. Never change what is inside the absolute value bars.just change the answer. Practice: Solve: 4 1 0 0 4
5 Solving Quadratic Equations ( Method) Method 1 By Factoring Solving a Quadratic Equation by Factoring 1. If necessary, rewrite the equation in the general form a +b+c=0, moving all terms to one side, thereby obtaining zero on the other side.. Factor completely.. Apply the zero-product principle, setting each factor containing a variable equal to zero. 4. Solve the equations in Step. 5. Check the solutions in the original equation. Practice: In 1-, Solve each quadratic equation by factoring. 1. 5 = 0. = 8 15 Method By Completing the Square Solving a Quadratic Equation by Completing the Square 1. If necessary, divide by a, the number in front of the -term must be 1.. Move c to the other side to isolate the binomial + b. b. Complete the square by adding to both sides of the equation. 4. Factor the trinomial side of the quadratic equation. 5. Apply the square root property and solve the remaining linear equations. Practice: Solve by completing the Square: + 6 + 7 = 0 Method By Using the Quadratic Formula The Quadratic Formula The solutions of a quadratic equation in general form a +b+c=0, with a 0, are given by the quadratic formula b b 4ac a Practice: Solve by using the Quadratic Formula: 4 = + 7 5
6 Radical Equations A radical equation is an equation in which the variable occurs in a square root, cube root, or any higher root. Solving Radical Equations Containing n th Roots 1. If necessary, arrange terms so that one radical is isolated on one side of the equation.. Raise both sides of the equation to the n th power to eliminate the n th root.. Solve the resulting equation. If this equation still contains radicals, repeat steps 1 and. 4. Check all proposed solutions in the original equation. Practice: Solve: Section P-9 Linear Inequalities and Absolute Value Inequalities Solving Linear Inequalities in One Variable When solving inequalities remember: Flip the inequality sign if you multiply or divide by a negative number. Practice: In 1-4, (a) Solve the inequality. (b) Use interval notation to epress each solution set. (c) Graph each solution set on a number line. 1. 5 + > 7 4. 1 > 14. < 9 4. 4 + 9 5 Solving a Compound Inequality When solving an and compound inequality: 1. get variable by itself between inequalities signs. and problems will graph as a connected line with no arrows Practice: In 1- Solve and use interval notation to epress solution sets. 1. 4 6 10 14. 10 5 8 6
7 Solving an Absolute Value Inequality When Solving Absolute Value Inequalities: If necessary, isolate absolute value epression on the left side of the inequality sign. Then, Write as two equations one with a positive answer and the other with a negative answer. Flip the inequality sign with the negative answer. o will be an and graph (no arrows---connected) o will be an or graph (arrows---oars) Practice: In 1-Solve and epress the solution set in interval notation. 1. 6 8. 5. 4 9 7
8 Geometry Review- More on Right Triangles 0 60 90 Triangle Theorem In Words: In a 0 60 90 triangle, the length of the is twice the length of the leg. The length of the longer leg is times the length of the leg. Hypotenuse = shorter leg Longer leg = shorter leg Symbols/Picture: Eample: Find the lengths of the missing sides of the triangle. 1.) 0 1.) 60 18 Eamples: Find the value of each variable. If your answer is not an integer, leave it in simplest radial form..) 4.) y 9 0 16 60 y 5.) 6.) 4 a b d a 14 60 c d 45 0 c 60 b 8