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.
Best Answer
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 $(-\infty, -6] \cup [6, \infty)$.