Flipping an inequality sign when multiplying by a negative number

Question

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.

Answer 1

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

Answer 2

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.

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

or

2. 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.