Flipping an inequality sign when multiplying by a negative number I came across a question that seemingly breaks the rule of flipping the sign when multiplying or dividing by a negative sign. Here is the equation to be solved.
$$|-x| \geq 6$$
The positive answer is as follows:
$$ -x \geq 6 .$$
Multiplying both sides gives:
$$ x \leq -6 .$$
However, I am confused by the negative answer.
I was able to get the correct answer by doing the following.
$$ -(-x) \geq 6, $$ which then becomes $$ x \geq 6.$$
However, if I apply the negative sign to the right side of the equation when taking the negative answer from the original equation, I get:
$$ -x \geq -6, $$ multiplying that by -1 would mean $$ x \leq 6,$$ which is not the correct answer.
I thought it may be that when I put the negative sign on the right side of the equation to create $$ -x \geq -6,$$ I need to flip the sign. However, that doesn't necessarily make sense as a rule since I don't need to flip the sign if I put the negative sign on the left side when I create $$-(-x) \geq 6.$$
Is there a rule on this? I can't wrap my head around why this happens and am wondering if there's a rule that guides to the correct answer when solving inequalities with an absolute value.
Edit: Fixed error.
 A: You write:  "The positive answer is as follows: $-x \ge 6$."  I think the cause of your confusion is that you haven't said what "positive answer" means.  I think you are assuming it means that $x$ is positive, and that is your mistake.
Here is a version of your solution that spells out more fully what "positive answer" means:  We consider two cases.
Case 1:  $x \le 0$.  Then $-x \ge 0$, so $|-x| = -x$, and the inequality becomes $-x \ge 6$, which is equivalent to $x \le -6$.  So for $x \le 0$, the inequality is true if and only if $x \le -6$.
Case 2:  $x > 0$.  Then $-x < 0$, so $|-x| = -(-x) = x$ and the inequality becomes $x \ge 6$.  So for $x > 0$, the inequality is true if and only if $x \ge 6$.
Combining the two cases, we conclude that the solution set of the inequality is $(-\infty, -6] \cup [6, \infty)$.
A: Okay.  The "positive" answer refers to the thing inside $||$ being positive.  If the thing inside $||$ is $-x$ then that is assumed to be positive is $-x$.  If $-x$ is positive than $x$ is negative.
So if we assume $-x \ge 0$ that is the same thing as assume $x \le 0$.
We get $-x \ge 6$.  Multiply by $-1$ AND FLIP!!!! and we get
$x \le -6$.  Which is fine because we are assuming $x \le 0$.
This is a "negative" answer for $x$.... but is a "positive" answer for $-x$.
=======
"I was able to get the correct answer by doing the following.
−(−x)≥6,
which then becomes
x≥6."
But that is NOT the correct answer if we are assuming $-x \ge 0$.  In fact that is the WRONG answer if we are assuming $-x \ge 0$.
But it is the correct answer if we assume $-x \le 0$.
.....
Basically if we have $|M| \ge k > 0$ we have two choices.

*

*If $M \ge 0$ then $|M| = M \ge k$ and $M \ge k$ and that is HALF of the answer.

or


*If $M \le 0$ then $|M| = -M \ge k$ so $M \le -k$ and that is the other half of the answer

And the answer is $M \ge k$ OR $M \le -k$.
.....
In your case you are dealing with $M = -x$ so the two answers are:
$-x \ge 6$ and $x \le -6$ [that is the "positive" answer for $-x\ge 0$]
OR $-x \le -6$ and $x \ge 6$
[that is the "negative" answer for $-x \le 0$]
.......
If that is too complicated we can always use the fact that $|-M| = |M|$ always (whatever $M$ is).
So we have $|-x| = |x| \ge 6$.
So if $x \ge 0$ we have $x \ge 6$ (that is the "positive" answer for $x \ge 0$).  And if $x \le 0$ we have $-x \ge 6$ so $x\le -6$ (that is the "negative" answer for $x \le 0$).
The answer is $x \ge 6$ OR $x \le -6$.
....
Consistent answers.
....
Anyway the rule about flipping when multiplying by a negative.  It's very simple.  You ALWAYS flip the inequality when you multiply by a negative.  ALWAYS.  NO EXCEPTIONS.  EVER.
