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My Question: How do I know when to flip the symbol (during multiplication or division) as opposed to when to subtract/add in order to change the positions of the numbers and variables?

Observe: When I did problem #4):

$-162 < 18r$

I simply divided both sides by 18 and flipped the symbol to get $r> -9$.

HOWEVER, in problem #5):

$\texttt{92 is greater than or equal to -4p}:92\geq -4p$

I divided both sides by -4 and got the incorrect answer of P is less than or equal to -23 after I flipped the sign. The answer key told me I was wrong. After I googled this problem, I learned that I had to subtract 92 from both sides and then put 4p onto the opposite side to end up with 4p is greater than or equal to -92 and then solve. the correct answer was p is greater than or equal to -23.

My Question: Why did I have to solve it differently in two look-a-like problems? Is there a rule to this? How will I know when to do what?

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    $\begingroup$ You have not, in fact, flipped the inequality in the first example, though you have written it in a different order. The term containing $r$ is still on the "greater than" side; $r>-9$ is the same as $-9<r$. The simple answer is that we flip the sign when multiplying the inequality throughout by a negative number. $\endgroup$ – AmpleMimic Aug 9 '15 at 13:17
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For the inequality $92 \geq -4p$, it's fine to divide both sides by $-4$, so long as you flip the direction of the inequality ($\geq$ becomes $\leq$) because $-4$ is negative. This happens for the reason described by user21820 in their answer.

$$92 \geq -4p$$ $$\frac{92}{-4} \leq \frac{-4p}{-4}$$ $$-23 \leq p$$

It would be fine to leave the solution in this form, but if you want $p$ on the left-hand side, you can read the last line from right to left:

$$p \geq -23$$

Another approach, as you mention, is to add or subtract as necessary on both sides:

$$92 \geq -4p$$ $$92 - 92 + 4p \geq -4p - 92 + 4p$$ $$4p \geq -92$$

At this point you can divide both sides by $4$, and since $4$ is positive, the direction of the inequality remains the same, so again you'll get $p \geq -23$.

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    $\begingroup$ Yup and I added a paragraph on addition and subtraction that is used in the alternative approach as you have illustrated. $\endgroup$ – user21820 Aug 9 '15 at 14:12
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Multiplication is essentially a scaling (bigger/smaller). When you multiply real numbers by $2$, you are essentially scaling the real number line by two times, with centre of expansion at $0$. So $1$ goes to $2$, and $-3$ goes to $-6$. Note that the numbers have remained in order, in that if $a < b$, then $2a < 2b$. This is not the case for multiplication in general. When you imagine multiplying real numbers by $r$ where $r$ is decreasing from $1$ to $0$, it is like shrinking the real line from its original down to nothing. We can carry on intuitively with $r$ going negative, where clearly we expect the real line to expand again but now with every point on the opposite side. This is really what multiplying by a negative number is like in the real world. One can easily see that multiplying real numbers by any negative real number causes the numbers to now be in reverse order. Also, multiplying real numbers by $0$ causes the order to collapse. Division by a real number $r$ is simply undoing multiplication by $r$, which is possible only when $r \ne 0$ since multiplication by $0$ can never be undone! Again, it is clear that division by a positive real number keeps the numbers in order, whereas division by a negative real number causes them to reverse order.

Addition/subtraction is essentially a translation (shift). When you add $2$ to real numbers, you are essentially moving the real number line by two units in the positive direction. Adding $-2$ to real numbers is essentially moving the real number line by two units in the negative direction. Subtraction of $r$ is simply undoing addition of $r$, so subtracting $2$ is the same as adding $-2$, and subtracting $-2$ is the same as adding $2$. Note that whatever we add or subtract, the real numbers remain in the same order as before.

And by the way, you must really understand what you are doing in mathematics so that you actually can not only correctly interpret mathematical statements but also can justify them to other people. Otherwise mathematics will forever be a place governed by rules that you cannot grasp. If my above explanation is not sufficient to tell you the meaning of multiplication and ordering, ask away.

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