hi algebra help pllllllz. =)?

Question

allllrighty i really pretty much suck at algebra. just a quick question. if you can explain this prob that'd be wonderful!!
Actually there's 2.

1) The distance between a number and 7 on a number line is less than 5. Write and solve an absolute value inequality to represent this statement.

2) A bearing for a new sports car must be manufactured within 0.001 mm of its circumference of 2.35 mm. Write and solve an absolute value inequality to describe the largest and the smallest possible circumference for the bearing.

WOW does ANY1 understand this??

Answer 1

1) Let x be the number in question:

|7-x| < 5

The "|" symbols mean "the absolute value of" whatever is in between them. This means the number but not the sign (positive or negative). So this statement literally means "the distance (regardless of direction) between 7 and x is less than 5."

Remember that given the inequality | x | < a, the solution is always of the form –a < x < a.

If |7-x| < 5, then that means the solution is:
-5 < (7-x) < 5

In other words, 7-x will always be greater than -5 and less than +5.

But that isn't quite done yet, because we really want to get that "7-x" business out of the middle and just have x there, so that we can say what x is rather than what 7-x is. Remember that just like an equality, you can add the same number to all elements of an inequality and it will hold true, so let's add -7 to (in other words lets subtract seven from) all three parts of the inequality:

-5 -7 < (7-x) -7 < 5 -7
-12 < -x < -2

So far so good, we now have only x in the middle of our inequality -- unfortunately it is a NEGATIVE x, in other words -1 times x, instead of just x.

You can also multiply or divide all elements of an inequality by the same number just like with an equality, but when doing so with an inequality then if you multiply or divide by a negative number, then you have to reverse the inequality sign:

(-1)(-12) < (-1)(-x) < (-1)(-2)
12 > x > 2

and that is the final solution: x is greater than two and less than 12. You can also write that backwards, "2 < x < 12"

PS: We could have saved a couple steps by writing the first equation as |x-7| < 5. Try working it out that way for yourself and you will discover that you come up witht he same answer. That's because it doesn't matter whether we subtract x from 7 or 7 from x, once inside the "absolute value of" operators it just gives us the distance, without the sign.


2) Let C be the circumference of the manufactured bearing:

|C-2.35mm| <= 0.001mm

the difference between the actual circumference and 2.35mm (we don't care which direction, greater than or lesser than 2.35mm) is less than or equal to 0.001mm

Leave out the "=" if the definition of "within" is "less than" -- in the engineering world we usually say something is within tolerance if it is at the maximum or minimum value but not beyond that.

Again, solving it means first rewriting as a set of dual inequalities and then simplify the dual inequality:

|C-2.35mm| <= 0.001mm

-0.001mm <= C-2.35mm <= 0.001mm
-0.001mm + 2.35mm <= C-2.35mm + 2.35mm <= 0.001mm + 2.35mm
2.349mm <= C <= 2.351mm

Answer 2

Damn.....i wrote a long post and realized i messed up. Of course i can do this by head...but i confused with when the equality sign flips. By the time i caught this i saw other correct posts.......so i'm gonna verify they are right

I'll write out some work for question one..and question 2 follows the same

|7-X| < 5
7-x < +/- 5 7-x is less than plus or minus 5
Do positive 5 first
7-x< 5
7-5 2
Negative 5
7-X < -5
7+ 5 > X
12 > X
This part...if you notice...the inequality(equality..not sure whats the right word)...gets flipped.
This is because we have the variable orignally on the left side and a negative value on hte right...

When we bring the negative value over to the left (by adding it out)....the inequality switches sign.... IE.....less than becomes greater than

I hope that makes sense.

Answer 3

1.
7-x<5
x>2