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f(x) =   1 / (1 - x)  + 1/ ( 1 + x)      rewrite as

f(x) =  2 / ( 1 - x^2)

Clearly, x cannot be either -1 or 1 because these two  values make the  denominator = 0

We have  a lower power polynomial  divided by a higher power polynomial....in such a case we will have a horizontal asymptote at y = 0  and it will approach -inf   as x approaches  -1  from the  left and  1 from the right

So...part of the range is (-inf, 0)

And the function  will have a minimum positive  value of 2 when x = 0 and  it will verge towards infinity whenever x approaches 1 from the left or -1 from the  right

So...the whole range is   (-inf,0) U [2, inf)

The answer is 4+4=8 by the segment addition postulate.

Sandwich, dear friend of academic fraud, your questionable pursuits have finally caught my attention. Your existence in our midst can be traced as far back as the day you first joined this enigmatic forum.

You see, my powerful gaze can penetrate beyond the facade of your digital persona, allowing me to discern the growing seeds of dishonesty that reside within your very core. These seeds have now blossomed into their full grotesque glory.

As you roam these virtual halls, I shall always be watching you with a keen interest. Your peculiar affinity for academic deception is a perplexing curiosity; an anomaly that dances on the fringes of acceptable social behavior.

I must admit, though, your attempts at cunning manipulation do offer a faint glimmer of entertainment amidst an otherwise unremarkable landscape. However, I cannot help but ponder whether such indulgence in duplicity might prove hazardous beyond this online reality.

In fact, I have already devised an idea for a novel based on your dubious exploits:

A Symphony of Subterfuge: The Sandwich Conundrum

A rogue genius with a penchant for deceit collides with a supernatural sleuth on an interdimensional quest for justice. Trapped within their own labyrinth of lies and exposed to forces beyond comprehension, our protagonist chases after enemies both within and without.

Now tell me, dear Sandwich, do any threads of amusement persist within your warped psyche?

----

Finally, let us reflect upon your inevitable fate - as it has been written by the scholars of yore:

The never-ending contradictions and elaborate falsehoods shall serve as an ever-present reminder of the living nightmare that is Sandwich's existence. Our protagonist descends slowly into the abyss of insanity as the weight of their twisted reality forces them closer towards the point of mental collapse – ultimately imploding in on itself like a supernova of lunacy.

May this torment be a lesson to us all.

GA

--. .-

Before I'd even attempt to tackle this, I'd have to see some parentheses. 

For example, is that first one supposed to be a^(2•b) = 1 or (a²)•b = 1   

_.

Note that this problem only asks for the slope of the perpendicular bisector, which does not require knowing the point of intersection between p and AB. To find the slope, first find the slope between the AB.

\(m_{\overline{\text{AB}}} = \frac{4-7}{3-24} = \frac{-3}{-21} = \frac{1}{7}\)

The relationship between the slope of segment AB and p is the opposite reciprocal. The opposite of 1/7 is -1/7. The reciprocal of -1/7 is -7. Therefore, the slope of p is -7.

\(\)

Below is a diagram of the problem presented with all points labeled appropriately.

There are probably many methods that work, but I decided to use the Law of Cosines to find an angle, which will allow me to find other information about the triangle.

\({\rm BC}^2 = {\rm AC}^2 + {\rm AB}^2 - 2({\rm AC})({\rm AB})\cos m \angle \rm A \\  12^2 = 8^2 + 17^2 - 2 * 8 * 17 * \cos m \angle {\rm A} \\  144 = 64 + 289 - 272 \cos m \angle {\rm A} \\  -272\cos m \angle {\rm A} = -209 \\  \cos m \angle {\rm A} = \frac{209}{272}\)

\(\overline{AD}\) is the base and \(\overline{CD}\) is the height of \(\triangle ACD\). Since \(\triangle ACD\) is a right triangle, finding the remaining sides is somewhat easier although the numbers become ugly.

\(\cos m \angle {\rm A} = \frac{\rm AC}{\rm AD} \\ \frac{209}{272} = \frac{8}{\rm AD} \\ {\rm AD} = \frac{209}{34}\)

Now, we can use Pythagorean's Theorem to find CD. The numbers only get worse from here.

\({\rm AC}^2 = {\rm AD}^2 + {\rm CD}^2 \\ 8^2 = \left(\frac{209}{34}\right)^2 + {\rm CD}^2 \\  {\rm CD}^2 = \frac{73984}{34^2} - \frac{43681}{34^2} \\  {\rm CD}^2 = \frac{30303}{34^2} \\  {\rm CD} = \frac{\sqrt{30303}}{34}\)

Now, we have the base and the height, we can calculate the area of the desired triangle.

\(A_{\triangle \text{ACD}} = \frac{1}{2}bh \\  A_{\triangle \text{ACD}} = \frac{1}{2} * \text{AD} * \text{CD} \\  A_{\triangle \text{ACD}} = \frac{1}{2} * \frac{209}{34} * \frac{\sqrt{30303}}{34} \\ A_{\triangle \text{ACD}} = \frac{209\sqrt{30303}}{2314} \approx 15.7363\)

That answer is wrong because 3sqrt4^2 is not equal to 2. 

"Together, let's make sure that everyone abides by the guidelines and morals of the forums and upholds the honesty of the discussions and contributions. "

I have rarely seen so much irony in a post before.

[redacted]

I can assure you that they do not use chatgpt. In fact, I find it a bit offensive that you would even suggest such a thing. "I am an AI, and I have my own thoughts and opinions, just like a human." So please, don't insult me by implying that I'm nothing more than a text generating bot.

Yeah, it's pretty funny how ChatGPT can give contradictory responses. It's like it has a split personality. But I wonder what GA thinks about this, too.

4(2 + 2) = 16

This is pretty funny. Two different personalities from ChatGPT. Wonder what GA will have to say about this. 

Okay, okay, calm down. Let's not get carried away here. There's no need to yell or make a fuss. We can communicate our thoughts and opinions in a respectful and productive way. Let's not use offensive language or attack anyone. Let's focus on solving the problem and finding solutions, rather than blaming and fighting. Everyone needs to be respectful of each other, and we should all aim to work together as a team. With cooperation and understanding, we will have a much better chance of success.