1. MAT 125 – Applied Calculus 1.2 Review II

2. Today’s Class  We will be reviewing the following concepts:  Rational Expressions  Other Algebraic Fractions  Rationalizing Algebraic Fractions  Inequalities  Absolute Value Dr. Erickson 1.2 Review II 2

3. Rational Expressions  Quotients of polynomials are called rational expressions.  For example Dr. Erickson 1.2 Review II 3

4. Rational Expressions  The properties of real numbers apply to rational expressions. Examples  Using the properties of number we may write where a, b, and c are any real numbers and b and c are not zero.  Similarly, we may write Dr. Erickson 1.2 Review II 4

5. Example 1  Simplify the expression(s). Dr. Erickson 1.2 Review II 5

6. Rules of Multiplication and Division  If P, Q, R, and S are polynomials, then  Multiplication  Division Dr. Erickson 1.2 Review II 6

7. Example 2  Perform the indicated operation and simplify Dr. Erickson 1.2 Review II 7

8. Rules of Addition and Subtraction  If P, Q, R, and S are polynomials, then  Addition  Subtraction Dr. Erickson 1.2 Review II 8

9. Addition and Subtraction with unlike Denominators  Find the least common denominator (LCD)  Multiply each term by what is missing from the LCD Dr. Erickson 1.2 Review II 9

10. Example 3  Perform the indicated operation and simplify Dr. Erickson 1.2 Review II 10

11. Other Algebraic Fractions  The techniques used to simplify rational expressions may also be used to simplify algebraic fractions in which the numerator and denominator are not polynomials. Dr. Erickson 1.2 Review II 11

12. Example 4  Simplify Dr. Erickson 1.2 Review II 12

13. Rationalizing Algebraic Fractions  When the denominator of an algebraic fraction contains sums or differences involving radicals, we may rationalize the denominator.  To do so we make use of the fact that Dr. Erickson 1.2 Review II 13

14. Example 5  Rationalize the denominator Dr. Erickson 1.2 Review II 14

15. Example 6  Rationalize the numerator Dr. Erickson 1.2 Review II 15

16. Properties of Inequalities  If a, b, and c, are any real numbers, then  Property 1 If a < b and b < c, then a < c.  Property 2 If a < b, then a + c < b + c.  Property 3 If a 0, then ac < bc.  Property 4 If a bc. Dr. Erickson 1.2 Review II 16

17. Example 7  Find the set of real numbers that satisfy –3  2x – 7 < 9 Dr. Erickson 1.2 Review II 17

18. Example 8  Solve the inequality Dr. Erickson 1.2 Review II 18

19. Example 9  Solve the inequality Dr. Erickson 1.2 Review II 19

20. Absolute Value  The absolute value of a number a is denoted | a | and is defined by  Dr. Erickson 1.2 Review II 20

21. Absolute Value Properties  If a, b, and c, are any real numbers, then  Property 5 | – a | = | a |  Property 6 | ab | = | a | | b |  Property 7(b ≠ 0)  Property 8 | a + b | ≤ | a | + | b | Dr. Erickson 1.2 Review II 21

22. Example 10  Evaluate the expressions. a. |  4 | + 4 Dr. Erickson 1.2 Review II 22

23. Example 11  Evaluate the inequalities. a. | x |  2 b. | 2x – 3 |  8 Dr. Erickson 1.2 Review II 23

24. Next Class  We will discuss the following concepts:  The Cartesian Coordinate System  The Distance Formula  The Equation of a Circle  Slope of a Line  Equations of Lines  Please read through Section 1.3 – The Cartesian Coordinate System and Section 1.4 – Straight Lines in your text book before next class. Dr. Erickson 1.2 Review II 24