# Math Help - Word Problem

1. ## Word Problem

The specifications for machining a piece of metal state that it must be 12cm long within a 0.01 cm tolerance. What is the longest the piece is allowed to be ? What is the shortest? Using l to represent the length of the finished piece of metal, write an absolute value inequality that states these conditions.

I have recently studied about absolute value and inequality through some textbooks so I am still kind of shaky. Could you please explain step by step.

Thanks.

Vicky.

2. Originally Posted by Vicky1997
The specifications for machining a piece of metal state that it must be 12cm long within a 0.01 tolerance. What is the longest the piece is allowed to be ? What is the shortest? Using l to represent the length of the finished piece of metal, write an absolute value inequality that states these conditions.

I have recently studied about absolute value and inequality through some textbooks so I am still kind of shaky. Could you please explain step by step.

Thanks.

Vicky.
$12 \pm 0.01 \times 10^n$

0.01 has to have some kind of unit or a percentage

3. I am sorry I left out the unit. It was within 0.01 cm tolerance.
Could you please explain again?

Vicky.

4. Originally Posted by Vicky1997
I am sorry I left out the unit. It was within 0.01 cm tolerance.
Could you please explain again?

Vicky.
If there is 0.01cm it means there can be 12+0.01 or 12-0.01.

If we let the length of the metal be l then $11.99 < l < 12.01$

Can you convert that to an inequality?

5. Then would it be
l 12 - l l < 0.01

12 - l < 0.01
-l < -11.99
l > 11.99

-12 + l < 0.01
l < 12.01

11.99 < l < 12.01

I think i got it right. Thank you so much.

6. ## Absolute value

Hello Vicky
Originally Posted by Vicky1997
The specifications for machining a piece of metal state that it must be 12cm long within a 0.01 cm tolerance. What is the longest the piece is allowed to be ? What is the shortest? Using l to represent the length of the finished piece of metal, write an absolute value inequality that states these conditions.

I have recently studied about absolute value and inequality through some textbooks so I am still kind of shaky. Could you please explain step by step.

Thanks.

Vicky.
The absolute value of a number tells you the size of the number, while ignoring its sign. So, for example, the absolute value of both $5$ and $-5$ is $5$. We can use the absolute value $|...|$ notation and write this as $|5| = 5$ and $|-5| = 5$.

Using this notation can be very useful in a problem like the one you've been given, because it can be used to describe the difference between two numbers without having to worry which of the numbers is the bigger, and which is the smaller. So, for instance, the difference between numbers represented by $a$ and $b$ can be written $|a-b|$, and you don't need to worry if $b$ is bigger than $a$ (which would make $(a - b)$ negative) because the $|...|$ signs will turn it into a positive number anyway. E.g. the difference between $7$ and $9$ is $|7-9| = 2$.

So, in your problem, we know that the difference between $l$, the actual length, and $12$ has to be less than or equal to $0.01$. So we can simply write

$|l-12| \le 0.01$.

and it doesn't matter whether $l$ is bigger than
$12$ or smaller.

OK?