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 Dependent Variable

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# Solving Equations with Rational Expressions

 Solving Equations by Clearing Fractions • Recall how to solve an equation containing fractions. We found the LCD of all denominators and “cleared fractions” by multiplying both sides of the equation by the LCD. Solve for x and Check Solution Your Turn: Solve for x and Check Solution Solve for x and Check Solution Solve for x Graphically find all x such that f(x)=-12 a. Sketch the graph of f on graph paper. Label the zeros of f with their coordinates and the asymptotes of f with their equations. b. Add the graph of y = -12 to your plot and estimate the coordinates of where the graph of f intersects the graph of y = -12. c. Use the intersect utility on your calculator to find better approximations of the points where the graphs of f and y = -12 intersect. d. Solve the equation f(x) = 2 algebraically and compare your solutions to those found in part (c). Your Turn • Page 677: 1 and 2 Chapter 8 Section 1 Introduction to Radicals Start by Solving: x2 = a • Three cases: Define 5. The solution of x2=a are called "square root of a"• In the case a<0 ,the equation x2=a has no real solutions. • In the case a=0 ,the equation x2=a has one real solution ,namely x=0 • In the case a>0, the equation x2=a has two real solution , The notation calls for the positive square root of a ,that is ,the positive solution of  x2=a. The notation calls for the negative square root of a , that is , the negative equation solution of  x2=a. Examples • Solve the following graphically and algebraically: Higher Order Roots • Start by Solving: x3 = a Figure 2.The graph of y=x3 inter- sect the graph of y=a in exactly one place. Square Roots • The number c is a square root of a if Example: So, -5 is a square root of 25 Principal Square Root • The principal square root is a nonnegative number given by: • The negative square root is given by: Note! • For all real values of a Simplify The Square Root of a Square • For any real number a The principal square root of a2 is the absolute value of a Simplify Higher Ordered Roots • The value c the nth root of a if • The nth root of a number is denoted Examples, Use Graph or Table to Check The nth root of an • To simplify where a is any real • The value of when n is even • The value of when n is odd Definition of a Rational Exponent • The nth root of of a is the same as raising a to the power of 1/n • Also, given exists then Examples: Rewrite in radical notation or in rational exponents Simplify  Examples • Use the table (where possible) to determine if the following simplifications are correct Negative Exponents also 