MATH HHEELLPP!!!! absolute value "and" & "or" statements?

Question

i am sssssoooooooo confused!! and i have a test on this tomorow and i have no idea what to do cuz i cant find much help! these r some of the homework ones i cant get: (i'm using the brakets for the abslute value symbole) [x]-6>16 , [x+1]-7.8<6.2, [3x]+2<8, HELP!!!!! ASAP

Answer 1

The "and" and "or" statements occur when we consider two cases in the absolute value. I'll explain using your examples.

|x| - 6 > 16
|x| > 22

Two cases:
1. First, we consider the case if x ≥ 0. Then |x| = x, so our inequality becomes x > 22. We assumed x ≥ 0, and came up with x > 22, so both these must hold, meaning we use "and." Thus, x ≥ 0 and x > 22. You should see that logically, these combine to simply x > 22.

2. Second, consider x < 0, then |x| = -x, and -x > 22, or x < -22. Combine: x < 0 and x < -22. So, x < -22.

Final answer combines case 1 and 2 with "or." Thus, answer: x > 22 or x < -22.

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|x + 1| - 7.8 < 6.2
|x + 1| < 14

Two cases:
1. Consider x + 1 ≥ 0. That means x ≥ -1. In this case, |x + 1| = x + 1. Our inequality becomes x + 1 < 14, which simplifies to x < 13. Combine the two conditions, x ≥-1 and x < 13, which can also be written in the combined form of -1 ≤ x < 13.

2. Consider x + 1 < 0. Then x < -1, then |x + 1| = -(x + 1). Then -(x + 1) < 14 => x + 1 > -14 => x > -15. Combine: x < -1 and x > -15, or -15 < x < -1.

Final answer: -1 ≤ x < 13 or -15 < x < -1
But, note that the endpoint is the same. Reorder:
-15 < x < -1 or -1 ≤ x < 13
This can clearly be combined into one:
-15 < x < 13
That's the REAL final answer. Or, if you want to use "and," then write x > -15 and x < 13.

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This one I'll get you started, and you should be able to finish it using the examples above as a guide.

|3x| + 2 < 8
|3x| < 6

1. 3x ≥0 or x ≥ 0. Then |3x| = 3x, and 3x < 6, meaning x < 2.

2. 3x < 0, or x < 0. Then |3x| = -3x.

Answer 2

Okay what you do first is you get the absolute value on one side. So...... for |x|-6>16 ... add 6 to both sides and you get:
|x|>22

Once you have the absolute value on the left side, the tricky part comes in, you're supposed to make 2 DIFFERENT inequalities from it.

One inequality will be the same as the original inequality, |x|>22 except the absolute value is removed. So it's:
x>22

The second inequality, however, you switch the > sign and make the 22 negative. Therefore:
x<-22

The reason for this is because |x| can be either x or -x so you're supposed to address both possibilities.

Now you have x>22 and x<-22. Now take a look at the two. You have one that says x is greater than 22 and another that says x is less than -22. This is an OR because NO number can be both greater than 22 AND less than -22... a number can only be greater than 22 OR less than -22.

When you have something like x>1 and x<10 then that's where you use AND. That's because a number can be greater than 1 and less than 10.

Answer 3

|x|-6 > 16

Since |-x| = x where x is positive read the above equation as the value of x, when taken as positive results in the equation being true. So

x < -6 and x >6 solves the above equation.

|x+1|-7.8 < 6.2

Now think of |x+1| as being some number call it a

Then a < 6.2 + 7.8 = 14

Now put back |x+1| for a:

|x+1| < 14 If x = -15 the |x+1| = 14 so we know x has to be greater than -15 . So that's one part of the range of x.

If x = 13 then |x+1| = 14 so x has to be less than 13. Thus,

-15 < x < 13

Hope that helps