Weekly Math Problem! #32

Equation with Complex Fractions. Yikes! 👀 Yeah...so...I purposely chose the crazy looking equation below, because I thought it would be fun to solve. 

This week's problem is the last problem of 2020. 😩 Not to fret, I will definitely be back in 2021 with more problems to help your brain sharp. (Gonna take me a little holiday break. 😌)

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     How solve equations with fractions
        ✔️     How to deal with complex fractions

WMP! #32 wants us to...

Happy solving!

Check back on Friday, December 18th for the solution, which will be posted below ⬇️.

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✏️📓 Solution Time! 📓✏️

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So, here we are to solve the last problem of 2020--WMP! #32. Just the other day, I started my blog. The other day was 11 months ago...wow. Time ⏳ is going faster than Usain Bolt. 

When it comes to fractions, sometimes we have options. With a complex fraction, I like to make it less complex. How did I do that? By getting rid of the denominators. For the major fractions in the problem, their LCD (lest common denominator) is 12. So, I multiplied both sides of the equation by the LCD. Here is the result:

 

After the work above, I definitely ended up with a less complex equation. However, there is still one fraction left. No worries. I will just repeat the process of multiplying both sides of the equation by the LCD. This time, the LCD is 2. 

L👀kie here.... after all of that work, the answer is zero. When was the last time you did that much work for nothing?! 😆 I did crack myself up...just a little bit. 

▪️ How do you feel about fractions??

▪️ Comment below with your responses and let me know what you thought about this week's problem.

Thanks for solving with me this week!

As a matter of fact...

THANK YOU FOR SOLVING WITH ME THIS YEAR❗️

WMP! #33 is up next and will make it's appearance on Sunday, January 10th, 2021.

Cheers!

The Younge Lady

Weekly Math Problem! #31

Absolute Value Inequality. I checked my list of problems from this year, looking to see what topic I hadn't posted yet. There are many, but I have settled on posting an absolute value inequality this week. Just keeping things mixed-up. I find that this topic isn't too bad. 

When working with students who are learning this topic, I notice that the most common mistake they make is only solving one inequality to get their final answer. Students get frustrated when points are lost and ask why, if the answer is right. I gently remind them that the answer is right but it's also incomplete. "You're missing the second answer." If a student remembers the method for solving absolute value inequalities, then they can most often make a mistake in the solving...especially if not comfortable with fractions. 

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     How to add/subtract fractions
        ✔️     The definition of absolute value and how to apply it
        ✔️     How to solve linear inequalities

WMP! #31 wants us to...

Happy solving!

Check back on Friday, December 11th for the solution, which will be posted below ⬇️.

Shameless 🔌 Plug: Follow me on Instagram @TheYoungeLady
Buy Me a ☕️ Coffee: TheYoungeLady

✏️📓 Solution Time! 📓✏️

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Here is how I went about solving WMP! #31:

The way I solved it, probably, isn't common, but it sure works. 

▪️ How did you solve the inequality above??

▪️ Comment below with your responses and let me know what you thought about this week's problem.

Thanks for solving with me this week!

Up next...WMP! #32

😄➡️

Cheers!

The Younge Lady

Weekly Math Problem! #30

De Moivre's Theorem. Sometimes, when I'm thinking about what problem to post for the week, topics just come to mind. Somehow, DeMoivre and his theorem was this week's winner. 🎉 The last time I really encountered this topic was well over five years ago...way back when I took a precalculus course. Now that I think about it, 🤔 I don't recall going over this topic in any tutoring sessions over the years. But then again, I've been tutoring for a long time and may not really remember if I actually did or not. 🤷🏿

I would also like to say the following: When I learned this topic, I was enrolled in a summer school course--about four weeks long. I learned the topic quickly without knowing what the practical applications of the theorem is. It felt like the topic was being learned for the sake of learning it, without any real-world connections being made. Even though I didn't ask the question then, I can ask it now and find an answer somewhere on this internet. Okay, I am done with that small rant. 

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     How to convert complex numbers to polar form
        ✔️     How to apply DeMoivre's Theorem to find powers of complex numbers
        ✔️     How to evaluate trigonometric functions

WMP! #30 wants us to...

Happy solving!

Check back on Friday, December 4th for the solution, which will be posted below ⬇️.

Shameless 🔌 Plug: Follow me on Instagram @TheYoungeLady
Buy Me a ☕️ Coffee: TheYoungeLady

✏️📓 Solution Time! 📓✏️

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Lowkey, I was a little annoyed by the question, after I began solving it. 😒 I didn't realize the numbers were gonna be this crazy. 😅  As soon as I saw 520 ...👀...smh...no words. I realized it was too late to turn back now, so I pushed forward to finish. I will say this, I was getting nervous about whether or not I was correct and needed to verify my numbers. Oh the anxiety I was having. 

Before applying De Moivre's Theorem, some preliminary calculations need to be performed.

Notice, I rounded the angle, 𝛳, in degrees to the nearest thousandth. I didn't want use radians because I am more comfortable with degrees. Next up, the application of De Moivre's Theorem.

Do you see what I mean about the numbers being crazy? 🤪 ...in the trillions! I am aware that answer above is not as accurate as it should be, due to my rounding of 𝛳 to the nearest thousandth. So, I have prepared a more accurate calculation below. 

My final thought: I didn't find the problem difficult to execute. Anxiety was triggered in me when I saw the first large number in my calculation. The anxiety lead me to begin doubting my mathematical abilities. Even though I am good at math, I still very much understand, what it feels like to be fearful when doing it.

▪️ What kind of emotions do you feel when doing math??

▪️ Comment below with your responses and let me know what you thought about this week's problem.

Thanks for solving with me this week!

Up next...WMP! #31

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Cheers!

The Younge Lady

Weekly Math Problem! #29

Logarithmic Equation. Y'all, it has been a loooong weekend for me. I...am...tired. 😩 Hopefully, your weekend went better than mine did. Because I'm tired and this week is a holiday week, this week's problem is a short one. Plus, I will be posting the solution earlier than usual. 

I don't encounter logarithmic equations a lot, but when I do, my eyes start to begin identifying which logarithm rules I need to use. To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     Logarithm rules
        ✔️     How to solve a quadratic equation

WMP! #29 wants us to...

Happy solving!

Check back on Wednesday, November 25th for the solution, which will be posted below ⬇️.

Shameless 🔌 Plug: Follow me on Instagram @TheYoungeLady
Buy Me a ☕️ Coffee: TheYoungeLady

✏️📓 Solution Time! 📓✏️

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I won't delay. Here's how I solved the equation...

Just in case, you're wondering why x = -1 was rejected, I'll briefly explain. The domain for a logarithmic function does not include negative values. There is no exponent--positive, zero or negative) that will yield a negative result. 

Just to ensure that x = 4 won't be rejected, check it:

So, this equation has one solution, x = 4.

▪️ Comment below with your responses and let me know what you thought about this week's problem.

Thanks for solving with me this week!

Up next...WMP! #30

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Have an amazing Thanksgiving!

Cheers!

The Younge Lady

Weekly Math Problem! #28

Two-Variable Nonlinear System. One of my goals with this blog is to challenge myself to solve/do problems that I'm not used to. Over the years, I've solved many linear systems of equation using a the three methods--graphing, substitution, and elimination. I haven't solved nearly as many nonlinear systems, especially ones that look like the one we're solving this week. However, after examining the problem, I know how I want to algebraically tackle this question. C'mon and tackle with me...

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     Substitution method
        ✔️     Elimination method
        ✔️     How to solve a proportion
        ✔️     How to solve a quadratic equation

WMP! #28 wants us to...

Happy solving!

Check back on Friday, November 20th for the solution, which will be posted below ⬇️.

Shameless 🔌 Plug: Follow me on Instagram @TheYoungeLady
Buy Me a ☕️ Coffee: TheYoungeLady

✏️📓 Solution Time! 📓✏️

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As mentioned before, I knew how I wanted to algebraically tackle solving this system after examining it for a bit. Okay, okay....who am I kidding? I kind of did both. 😊 L👀k...

So, the solution to this system has four points for the solution--{(1, 1/3), (1, -1/3), (-1, 1/3), (-1, -1/3)}. However, I was still curious to see what the substitution method looked like. There is more than one way to apply the substitution method, so here's how I did it... 

As you can see, solving for y using the substitution method the way I did, does yield the same values for y I got with the elimination method. Whew! If I continued to solve for x with the substitution method, I will also get the same values as I did above. 

Now, you know I couldn't end this problem without having a graph of the system. The graphs have been labeled with an "A" and "B" in accordance with how I labeled them above.  

**This plot was created using Geogebra's Graphing Gaclulator.

▪️ What method did you use to solve this nonlinear system??

▪️ Comment below with your responses and let me know what you thought about this week's problem.

Thanks for solving with me this week!

Up next...WMP! #29

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Cheers!

The Younge Lady

Weekly Math Problem! #27

Hyperbolic Functions. So....about hyperbolic functions...👀 They're not things that I have worked with a lot, however, they are similar enough to trigonometric functions that I don't feel too intimated to work with them. I've had the opportunity to work with them more this semester because of tutoring. So I though it would be fun to have them as part of this week's WMP!

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️     Differentiation rules
        ✔️     Pythagorean identities
        ✔️     Factoring techniques

WMP! #27 wants us to...

Happy solving!

Check back on Friday, November 13th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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Differentiation time! The first thing I do is rewrite the function in a form that allows me to take the derivative more comfortably. What do I mean by that? Well, the second term in the function is rational, so one may be inclined to use the quotient rule to take its derivative. However, I am not so inclined. 🤫 I try to avoid the quotient rule whenever I can. 🤫

By rewriting the second term with a negative exponent, I can take the derivative of it using the power rule. Now, that that's complete, let's get to the taking the derivative of the function.

That's the derivative. It has been taken and simplified, ever so slightly, after applying the applicable differentiation rules. ...But I'm not going to leave my answer like this. Nope! There is more simplification that can be done. Here goes:

These are the kind of problems professors like to give. 👀 Let me tell you why. Are you 👂listening? Good. There was more work involved in simplifying the derivative compared to the work needed to find the derivative. In general, points are often lost in the simplification process. In my opinion, the differentiation process in this week's problem wasn't too bad. I or anyone could easily have made an error in the simplification process. How?

• By not knowing you can split the second term in the derivative up as a product of two factors, where one of them can be rewritten as a single hyperbolic function--tanh(x). 
• By not knowing you could factor the derivative. 
• By not knowing/being familiar with Pythagorean identities which allow for the derivative to be further simplified.

In a case like this, and lots of other problems, the algebra is where students make mistakes...not the calculus.

 

▪️ What is your relationship with hyperbolic functions like??

▪️ How did you do this week's problem??

▪️ Comment below with your responses and let me know what you thought about this week's problem.

Thanks for solving with me this week!

Up next...WMP! #28

😁➡️

Cheers!

The Younge Lady

Weekly Math Problem! #26

Logic. A recent tutoring appointment is the cause of me reminiscing back to the time in my life when I learned logic. At that time, it was a topic that would appear on the Sequential Mathematics II Regents examination I would take. These days, logic is not part of the current high school math curriculum. Here's the funny thing: I wasn't tutoring math when this topic came up; I was tutoring a student in philosophy. 🤷🏿 Now, I am by no means a philosophy tutor. However, I was able to help a student with logic which is a major part of the philosophy course the student came to get help with. During one of our sessions, I described to the student how I learned some of these logic topics as part of my high school math curriculum in New York. So I just had to look it up and make it a problem of the week. 🤓 

The problem below is taken from the January 1998 Sequential II Math Regents, Part III, Question 41, worth 10 points. These kinds of questions were like puzzles to me. 

To solve this week's problem in completion, you need to recall the following math skills:

        ✔️ Logic rules of inference
        ✔️ How to construct a logic proof

Ready? Here is WMP! #26:

Happy solving!

Check back on Friday, November 6th for the solution, which will be posted below ⬇️.

Shameless plug: Follow me on Instagram @TheYoungeLady

✏️📓 Solution Time! 📓✏️

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Alright, let's get this proof going. First thing's first...symbolization.

Now, that we can see the symbols, it'll be easier to compare them to the rules of inference which will be used to help us prove the proposed conclusion. Here is how I completed the proof:

In the proof, I used "Law of Disjunctive Inference" to refer to Disjunctive Syllogism (or Elimination). Why? I'm kinda old school and that's how I remember it. 👵🏿  

 

▪️ Have you ever done these types of proofs??

▪️ If you have, did you do them in math or philosophy??

▪️ Comment below with your responses and let me know what you thought about this week's problem.

Thanks for solving with me this week!

See you next week for WMP! #27

😁➡️

Cheers!

The Younge Lady