Absolute value inequality...?

Question

l 3x - 7 l >/= -3


How do I solve this? To me...it may as well be chinese!!!

Answer 1

Absolute value is always greater than zero. Imagine all numbers on a number line. Absolute value is the distance from zero. So for example, 3 and -3 are both 3 numbers from zero, so they both have an absolute value of 3.
Long story short, any absolute value is greater than any negative number. In your question, the answer would be "all x"

Answer 2

A.) Are you SURE that you have stated this problem CORRECTLY?

I ask, because - 3 is a NEGATIVE number. Any "modulus" is necessarily >/= -3; so that NO RESTRICTION whatsoever is placed on x by this equation:

==> ' x ' can be ANYTHING !

B.) However, let us suppose that the problem were this:

"Solve the equation | 3 x - 7 | >/= + 3."

I shall now show you how to do this VERY SIMPLY, and without bothering about the things you'd have to do if you followed the advice that I criticized in another "modulus" question that you asked earlier.

(Another problem with that BAD advice is that if the equation becomes, instead, an INEQUALITY, you have to SWITCH THE SIGN of the inequality when you change the sign of stuff outside the "mod" sign. You will see, if you check 'Taranto's' solution of this amended problem, that THAT'S what he did. With that method, you simply have to stop too many times in midstream and think "Now, is this right, or do I have to rethink something?" I'd rather have a simple method that is guaranteed to work every time, without second thoughts part way through. Who wants to deal with that other arbitrary CRAP when it can all be avoided very simply ?!)

O.K., so in the amended problem, we have two simple possibilities:

(1) (3x - 7) is +ve: then (3 x - 7) >/= + 3. So 3 x >/= 10, or x >/= 10/3.

(2) (3x - 7) is -ve: then (7 - 3 x) >/= + 3. So 4 >/= 3 x, or x
Don't you SEE how simple that was? Just two lines. No messing around saying to yourself "Now let me see, if the absolute value of this is greater than that, but on the other hand if the expression itself is actually negative that means that the expression is actually less than the negative of that ... ." I mean, really, life is short. WHY are people SO RESISTANT to doing things in a much more natural and straightforward way that doesn't require all these mathematical "ifs, ands, buts, second thoughts and 'on the other hands'."

I hope the simplicity of separately considering the possibilites that the expression is either (a) positive, or (b) negative, will now have impressed itself upon you !

Live long and prosper.

Answer 3

It is true for all values of X.

The left hand side of the inequality says "Multiply X times three and subtract seven -- then change the sign if it is negative." The two upright bars mean "take the absolute value of what is inside" -- where the absolute value is always positive. So, |5| = 5 and |-5| = 5.

Since the left hand side is always positive, it is always greater than or equal to -3.

This doesn't really tell you how to solve it, though. Let's change the problem. Suppose it was

l 3x - 7 l >= 3

This would be true whenever

3x - 7 >= 3

or 3x >= 10
or x >= 10/3

But it would also be truen whenever

3x - 7 <= -3

This is true -- because if 3x - 7 <= -3 -- then its absolute value is greater than three.

or 3x <= 4
or x <= 4/3