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Hint

1

what is a undefined or restricted value?

a value that makes the denominator = 0

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2

How do we find restricted values?

Set the denominator equal to 0 and solve

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3

Find the restricted value:

3x/x+5

x = -5

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4

Steps to simplify rationals

Factor both the numerator & denominator

cancel

**like factors**from the numerator & denominator**write simplified answer**

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5

Simplify.

__xÂ˛ + 3x__

xÂ˛ + 5x

x + 3 / x + 5

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6

Simplify.

x(2x-3)

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7

Simplify.

2 / x + 3

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8

Steps to **multiply rationals**

**DO NOT MULTIPLY. I REPEAT, DO. NOT. MULTIPLY****Factor numerators & denominators****cancel like factors and write remaining expression**

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9

Steps to **divide rationals**

**DO NOT DIVIDE.**multiply by the reciprocal of the second rational expression

Follow process for multiplying

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10

**Multiply.**

(x + 2) (x - 1) / x(x + 1)

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11

**Divide.**

(x - 8) ( x - 5) / (x + 7)Â˛

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12

Steps to **add rationals**

**ALWAYS**make sure the denominators are the sameThen, add the numerators by

**combining like terms.**

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13

Steps to **subtract rationals**

Make sure the denominators are the same

Combine like terms, and donâ€™t forget to distribute the negative sign if necessary!

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14

To find **common denominators**â€¦

Only use each factor once

factor each denominator individually & find the

**LCD**multiply them together to get a combined denominator

multiply the numerator by the respective denominator (see example)

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15

**Add. **

3x + 2 / (x + 2)(x - 2)

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16

**Subtract.**

-4 / x + 2

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17

Steps to **solving rational equations**

find common denominators (multiply rationals by a factor to make them all have the same denominator)

cross out the denominators and solve the remaining equation

check for extraneous (restricted values) solutions

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18

**Solve.**

x = -1

4 is extraneous.

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19

vertical asymptote

set denominator = 0 and solve.

written as

**x = #**

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20

domain

All real numbers **except the vertical asymptote you found.**

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21

horizontal asymptote

Use the degrees of the numerator & denominator

equal degrees : coefficients of leading terms

numerator < <

*denominator*: HA = 0> > denominator :*numerator***NO HA**

Written as y = #

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22

range

All real numbers **except the horizontal asymtote you found**. If no HA, then its all reals.

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23

x intercept

set **numerator = 0 **and solve

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24

y intercept

**plug in zero** for all x-values

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25

**holes**

if you can cancel a factor from the numerator & denominator, then there is a **hole** at that x value.

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26

Find the characteristics:

x - 2 / x + 2

**factored**VA:

**x = -2**Domain:

*R: x cant be -2*HA:

**y = 1**Range:

*R: x cant be 1*X-INT: (2,0)

Y-INT: (0,-1)

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