Question

graph/ solve

Answer 1

we have negative numbers with a relation
so *we have to be careful*

the "fast way" is multiply both sides by -1 and *switch the relation operator*

the slow (but easy to do) way is add +3 to both sides
then add +3 | x-5| to both sides

either way, what do you get?

Answer 2

0

Answer 3

/x-5/<1

Answer 4

do it in steps
-3 | x-5| < -3

which way do you want to do it? fast way or slow way?

Answer 5

fast way of course!

Answer 6

fast way:
multiply both sides by -1
*flip the relation operator*

Answer 7

would it be 3/x-5/<3 ??

Answer 8

you did step 1 ok.
but step 2 means you change < to >
or if you had > you would change it to <
or \(\le\) would be changed to \( \ge\)
that is the "flip" part

Answer 9

3/x-5/>3

Answer 10

yes. the problem with the fast way is you have to remember to flip the operator... if you don't have a good memory for "details", the slow way is safer (one less rule to remember)

so now you have
3 | x-5| > 3

it is safe to divide both sides by 3. you get
| x-5 | > 1

Answer 11

with absolute value problems we *always* have two conditions
either x-5 is positive , in which case , drop the | |
or x-5 is negative, in which case we replace the | | with -1*(x-5)

Answer 12

let's do the x-5 is positive condition.
we have
x-5 > 1

now add +5 to both sides

Answer 13

x>6

Answer 14

now do the x-5 is negative case. can you do that one?

Answer 15

yes(hopefully)!
So i got x<4

Answer 16

looks good.
x<4 or x>6
are the solutions
put an open circle at 4. the open (not solid , filled in) means 4 is not part of the answer, but we can get as close to 4 as we want (without being exactly 4)
and x smaller than 4. all numbers smaller (to the left) of 4 are part of the solution
as it would take a long time to draw a line showing all of the numbers, we use an arrow tip to show the answer includes all numbers the arrow points towards.