D)

ABSOLUTE V?\LI]E:

Note:

lxl>0

always

x-2,

tf D2

Example:

l*-21

-x*2, if x<2

Geometric interpretation of absolute

value:

Note:

lxl

gives

the

distanrce

x is from

0

on

the x-a:ris. Note:

lxflx-01

Examples:

lE:2

glves

solutions

2

&,-2,

l4<2

grves

allnumberstn(-2;2),

lx-2p3

gives

solutionS

-1

&

5,

lx+l1<2

gives

all

nr:mbers in

(-3;1)

Note:

Can't

use distance

interpretation if the

coefficient

of

x is not 1

i.e.,

l2x-11:2,

l7x-31<5,

l5x+11>2,

So we need special

procedures

to solve tlrese

PROCEDURES:

FORM

#1)

lax+bFs

-)

ax*f:g

OR

s;s*fo:

-c

Example:

l2x-3[5

-+

2*3:5

QR

2x-3:

-5

-+

x:4

OR.

x: -1

FORM

#2)

lax+bl>c

+ il(+b>c Of,. ax+t<

-c

Example:

l5x+2p3

+ 5x+2>3

OB

5x+2<

-3

-+

p1i5

OR

x<

-1

FORM

#3)

la:<+bl<c

+

-c<a>(+b(c

+

-c<ax+b

AND ax*b<c

(both

must be satisfied)

Example:

l2-7xl<5

+

-5<2-7x<5

-+

-5<2-7x

4Ie

Z-7x<5 + 7x<7

4IA

4<7x

-+x<1

AND -317

<x-

This

gives

the

interval (-317;T)

on

the x-axis.

I

Note: Basic definition:

lxl

:

I

if x>0

if x<0

)

f

*-2, if x-2>O

:{

l-x+2,

if x-2<0