Solving an Inequality: |5 - x| <= 4

[tristatefabricatorsinc]

I have not done inequalities for some time... can someone give me some hints as to the steps involved for solving a problem of this nature?

|5 - x| <= 4

I am guessing the answer must be a range...

[stapel]

You are correct about the nature of the solution. See if this example helps at all:

. . . . .| x | < 5

. . . . .-5 < x < 5

Do you see (above) how to get rid of the absolute-value bars?

Eliz.

[tristatefabricatorsinc]

I kinda understand it, but not very well. I know a negative is less than a positive of course and the absolute value of a number is always positive. Can you give me some more samples or do a problem similar to mine with different numbers.

Thank You Very Much!

[tristatefabricatorsinc]

Would this be correct?

|5 - x| <= 4
-4 <= 5 - x <= 4
-1 <= -x <= 9

Answer:
9 <= x <= 1

[tkhunny]

I don't really like that method, since it works only for contiguous regions.

I prefer to be very deliberate about it.

When 5-x >= 0, |5-x| = 5-x
When 5-x < 0, |5-x| = -(5-x) = x-5

5-x >= 0 requires x <= 5
5-x < 0 requires x > 5

Then,

For x <= 5, solve 5-x <= 4
For x > 5, solve x-5 <= 4

Solution

For x <= 5, 5-x <= 4 or x >= 1 giving the interval [1,5]
For x > 5, x-5 <= 4 or x <= 9 giving the interval (5,9]

Complete Solution: [1,9]

You are almost there, except that you seem to have added 5. Try subtraction.