NOTES: Absolute Value Equations and Inequalities

Define absolute value (geometrically):_----'dc:.t=...::.=..' =t4..=..••=4=~'-)---Tf~'--"-...:..:...:::-.:.-~7F==:....------V rr---

Graph a distance of 5 units from the origin, Then write the equation using an absolute value sign .

•-5 oGraph a distance less than 5 units from the origin. Then write the inequality using an absolute value sign.

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Graph a distance more than or equal to 5 units from the origin. Then write the inequality using an absolutevalue sign. I

rX~5"I'lL (~S tK~5=>X£::S• I •

-~ 0 ~

Graph the following. Then write the answer in set-builder notation,

1.) Ixl= 3 ••• • I • • {~/'X;3~ ?G :-31-3 0 3

X.=:3 tn.-x=3

[?c= 3 '"""' )(-==--3]

2.) Ixl ~2 ••• • I • • [ "K.! - o? ~ 1G ~ ;2.1_-2- 0 2-

X ~::l ~-1C ~,;I-

~=~~ 1C~-i13.) Ixl>6 ..•( ~ , ~ It> [. -K.{ JC L-fo ,;n.. 1£>c, 1

~ ~" O't.. -x ).~ -(p 0 ~

0t .)."D'\.... XL-']

4.) Ixl<-2 ••• •XL-..l ~ -,c L-2.

(X~-l. ~ 1C > :t..J

5.) Ixl >-2 ~ ). IRX~ -2.. 6'"\.... -"}C ~-2..

l'}C~-l. I>'\.. ?C "" :L J6.) Ix+31< 2 .••I I" J I I • I • [1Z{ -SL..1C~ -13

-5" -( I'k:. +-36:l. ~ -(X+-3) ~:Ll1L..f.-3 L.2 a-...L ~+-3 ~ -2-1

Define absolute value (algebraically).'If a: ~ 0 --/L.-... I 0- / "C tl- .

-fI a. co -.4. ~ I CLI c -a....Solve ttie following. Graph your final answer and write your answer in set-builder notation.

1.) 4x-2::O;x+l::O;3x+4

4x-2. = 1(;+-1 ~

3'X-2..f: I3x~3x~, ()

-1C+- I 1:: 3 x+-L/-.2.X ~ 3

X ~-~

2.) 4x-8>6x+5 or 5x-8<-2ex L.. G,

U 1C£. ~s:

'ED'Is

?('; --iII •...---.l •..--~,.t~{-%. 6. ~ ~ 11 f 1(. I 1t:~ '"/s-J-o/~ I [-%. I] c- oo ) "Is-)

2 2 2 2 I 31 313.) --::O;-x--<- 4.) 2x--<-- or 2x-->-

15 3 5 15 4 10 4 10

7.) IX-41 ~ 2

"X-4 ~ ~~~(,

•• • ) >2- b

t.~(1c:b2. U)(~b3( - 00 j:t] u [" I 00 )

9.) 6+ 41xl < 2 + 61xl-2{x.\L -4

I}(\~ ~ ~ ~.).l.. ()"\.,-,c~2..)C ~-~

4;_! '!.-) {Xl 1CL.-7- u x> 2.5

-I

~~ [- ('-XL -;3(-, I 7/3 )

10.)

12.) \4x\+1>27 Ii/x l ~ ~~41L ~.2' g-1- -(4'Jc) ~ ~c,

4lC L -01'1::::' I~ 1{ c: - (-Y2..

<: eo)- '7l. (7"2-

[1C { ~ L-I3/~~ u 1C~ (~3I I (-00 -l3A J U ((3/ oe)5x+2 <28+3x / 1.. /~/

5X+ 2.. L.l. 8 +3 'X- o.-"..".t.. - (sx+--2}:!· J.V+-3k£)(1-2. ~ -2.8-3A

e» ~ -.30K ~ -/~

>

11 .) Ix + 2\ > 4

1(+2. > tI n... 1C4-.:2.. L -~--

[ 1C ( 1l L. - ~ u ~~2.3(-00) -,) U (2.../-XJ)

1 1--lx+91~-x-15 14.)2 213.)

-:t;+--q f: -~ +-.3o ~ - ( X'+-<1) 6 -].C+- 3 6

~Xl L. .:l. I ~~ q ~ 1(" - 30

t"~ 02~ 9 ~-30 c: (D tB_---,-_~ ~ ~ -t~ ( 3< .2-y. ;:=\ ['k-I1c f:. zYl-3 - [1L I - IS/if L 1C L 133 (-{J.7'1J (3)

15.) The Vikings play 36 games this year. At midseason, they had won16 games. How many ofthe remaining games must they win in order to win at least 80% of all their games?

~.:= ~ ~ ~""":1 .:, (J"''''

1:-+.-I'~ . g(3~)

1c ~ I~. ~ v~ ht.--J- ~ ~,I, •• ~

/3 ht--t-c..-- ~.

16.) The city parking lot charges $2.50 for the first hour and $0.25 for each additional hour. If themost you can pay is $6.50, how long can you park your car?