Solving inequalities!!!...HELP!!?

Question

Find the absolute inequality corresponding to x <= -22/3 or x >= 11/3

can you show me the steps so that i know how to solve this kind of problem...THANKS :)

Answer 1

This is a very confusing question. Are you certain that you have the inequality signs written correctly?

For x to be &gt;= 11/3, x must be greater or equal to 11/3 or x &gt;= 3 2/3. For x to be &lt;= -22/ 3, x must be less than or equal to -7 1/3. Combining the two inequalities, all values of x that satisfy one or the other of the inequalitites are in those ranges.

The solution is two ranges of x. One range consists of all values above (and including) 3 2/3 and the other consists of all values below (and including) - 7 1/3.

For all values between (but not including) -7 1/3 and 3 2/3, neither of the inequalities are met. I don&#039;t know of a specific notation to show that particular range as not meeting the ineaualities.

Answer 2

I&#039;m just guessing, but it looks like you want to transform this into

  |expression in x| &gt;= a number.

I assume that&#039;s what you mean by an absolute inequality.

Seems to me that if we modify those two inequalities so they have the same absolute value on the right, we&#039;re there.  I think the trick is to find a constant k such that

x + k &lt;= -22/3 + k

x + k &gt;= 11/3 + k (these are the original inequalities modified)

AND

11/3 + k = -(-22/3 + k)

11/3 + k = 22/3 - k

2k = 11/3

k = 11/6

x+11/6 &lt;= -22/3 + 11/6  = -33/6 = -11/2

x+11/6 &gt;= 11/3 + 11/6 = 336 = 11/2

so

| x + 11/6 | &gt;= 11/2

is a single inequality that encompasses both the original inequalities.