MAT 125 – Applied Calculus 1.2 Review II

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

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

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

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

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

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

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

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

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

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

Example 4  Simplify Dr. Erickson 1.2 Review II 12

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

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

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

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

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

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

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

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

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

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

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

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