Chapter 8 Rational Expressions, Equations, and Inequalities and Other Functions Taught by: Nicky Chan, Aaron Hong, Alina Kim, William Qin, and Nathan Si

§8.1 - Multiplying and Dividing Rational Expressions

Examples for §8.1 Simplify *Find the GCF and reduce. 25 and 45 have a common factor of 5.

Examples for §8.1 Simplify. *Factor *Reduce when two of the same expression is present in The numerator and denominator

Multiplying and Dividing Fractions  Multiplying *reduce numerator and denominator *multiply straight across

More Complex Multiplying *Factor *Cross multiply and reduce numerator and denominator =1

Dividing Fractions  when dividing, multiply the reciprocal *FACTOR *Reduce x *Multiply Straight Across

§8.2 – Adding and Subtracting Rational Expressions  To add and subtract, you need a common denominator.  The common denominator is the least common multiple (LCM) of all the denominators.  The LCM of 18 and 28 is 252.

LCM… Multiply circled expressions

Add/Subtract.  Remember: Find LCM for the denominator.

§8.3 – Graphing Rational Functions

 Vertical Asymptote: -5 ||x+5=0; x=-5  Horizontal Asymptote: y=3 [y=ratio of the leading coefficients (3/1)]  MAKE A TABLE xy Asymptote

THE GRAPH (continued)

 Horizontal Asymptote: NONE (Numerator degree is greater than denominator)  Vertical Asymptote: NONE (Cancelled out; there will be a hole)

THE GRAPH

§8.4 – Direct, Inverse, and Joint Variation  Constant of Variation – The variable “k” is the constant of variation.  Each type of variation uses k to compare 2 situations.  Solve most problems using 2 steps. (Usually, you will need to find k first, and then find the variable asked for in the question.)

Formulas

Examples for §8.4  If y varies directly as x and y=-15 when x=5, then find y when x=3.  Formula: y=kx  First: y=-15 when x=5 *Plug into the formula  -15=k(5) SOLVE FOR K k=-3 Second: Find y when x=3 Use k=-3 Y=(-3)(3) Y=-9 ANSWER = y=9

Examples continued…

 Suppose a varies jointly as b and c. Find a when b=10 and c=5, if a=12 when c=-18 and b=3.  Use the formula y=kxz. In this case, a=y, k=k, x=b, and z=c.  First, solve for k.  12=k(3)(-8) 12=-24k k=-1/2  Next, solve for y, or a. a=(10)(5)(-1/2) ANSWER: a=-25

§8.5 – Classes of Functions

Direct Variation y=3x

Absolute Value Function y=|3x-2|

Constant Function y=3

Greatest Integer Function y=[[x-1]]

Identity Function y=x

§8.6 – Solving Rational Equations and Inequalities

Solving a rational equation *Solve by clearing the fraction. Multiply the entire problem by the LCM, which is 24(3-x).

Solving Inequalities  First, you find the excluded values.  Solve the related equation by changing the inequality into an equals sign.  Then put everything on a number line and test the sections.

Examples for Solving Equations 3x *Solve what’s left: 3+2=2(3x)

The Number Line Test points less than 0, between 0 and 5/6, and greater than 5/6… Let’s try x=-1, x=1/3 and x=1. The equation was: 1/3x + 2/9x = 2/3