calculus help?

Question

find all numbers x such that

lx-2l < 1/100

i started it but omg i couldnt get it.

i subtracted 1/100
but i dont think thats right

id be so gratefull if you help me.

Answer 1

-1/100 + 2 < x < 1/100 + 2

= (199/100) < x < (201/100)

Interval: (1.99, 2.01)

Answer 2

The trick is to first remove the absolute value from the equation. Since the absolute value makes the expression positive (even if it is negative) this will give you two equations to solve.

The first and easier one to see is simply going to be
x - 2 < (1/100) Add 2 to both sides.
x < 2 + (1/100) => x < 201/100 This is the first bound for x.

Next, since anything negative in the expression will be come positive due to the absolute value symbol, you also have that
-(x - 2) < 1/100 Carry out the negative sign.
-x + 2 < 1/100 Subtract 2 from both sides.
-x < (1/100) - 2 Simplify.
-x < -199/100 To get rid of the - signs, add x to both sides and also add 199/100 to both sides.
199/100 < x => x > 199/100. This is your second bound.

Putting it all together, you know that x > 1.99, and x < 2.01. (201/100 = 2.01, and 199/100 = 1.99) Since x must be more than 1.99 but less than 2.01, you can write it as 1.99< x <2.01
Hope that explanation makes sense to you.

Answer 3

lx-2l < 1/100
This can be rewritten as:
-1/100 < x-2 < 1/100
Now add 2 to each term of the inequality getting:
2-1/100 -1.99 < x < 2.01

Answer 4

-1/100 < (x-2) < 1/100
Then add 2
2-1/100 < x < 2+1/100

That means that x is either 1, either 2 if x is a natural number

Answer 5

2 - 1/100 < x < 2 + 1/100 =>1.99 < x < 2.01
----------
Ideas: -1/100 < x-2 < 1/100. Add 2 to each term.

Answer 6

x-2 is either non-negative or not. if non-negative, |x-2| = x-2. if not, |x-2| = -(x-2). so, either
x-2 < 1/100
x < 2.01
or
-(x-2) < 1/100
x-2 > -1/100
x > 1.99
so 1.99 < x < 2.01

shortcut is to go straight to
-1/100 < x-2 < 1/100
and then add 2 to all 3 parts.