Tier 3: Part 1 – SAT/ACT Test Preparation · Tier 3: Part 1 – SAT/ACT Test Preparation P1-Introduction ... P1-1 LESSON: The Real Number System ... 8. √-75 [C Complex, Not Real,
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1.2 A Message to the student regarding problems ............................................................................................ 6
of understanding, which math does. .................................................................................................................... 7
P1-1 LESSON: The Real Number System ................................................................................................................ 8
P1-10 LESSON: Coordinates in a Plane ................................................................................................................ 31
P1-Introduction A Message to the student regarding problems: As you know, the more you practice the better and faster you will get, just like in any game or sport. So these problems are for those of you who want to get better and better. If you solve the problems given in the Simmons book you will be OK and be able to pass the Quiz, and probably do pretty good on the SAT and ACT. But, the more you practice the better you’ll get. And, I find I enjoy math more the better I get. I have included some problems that should also help you understand the math behind them better and better. Some of these problems may be pretty challenging, although you do have enough knowledge to solve them if you think long enough. DO NOT get discouraged when you tackle a hard problem and struggle with it. THAT HAPPENS to all mathematicians. In fact, we really seem to be understanding math at a deeper level when we struggle with a problem. If you get frustrated, take a break. If you have others to work with help each other. You will learn a lot by teaching and explaining things to others. That is really how I learned math. I started “tutoring” some of my classmates when I was 15 years old. And, they helped me too. Learning math is like any sport, the more we practice and coach others the better we get. So, I hope you enjoy these additional problems. IF YOU FIND A MISTAKE in my answers, Bully for you. I’m pretty sure there will be some mistakes creeping in from time to time. Sometimes, I leave them in for you to find a little gem. It will help you build your confidence, too. Never believe something just because it is written in a textbook. Verify it with your own thinking and understanding as much as you can.
I can’t tell you how many “mistakes” I have found in various textbooks over the years. And, sometimes on tests, too. You might want to practice my Motto: “Never be Satisfied, Always be Content.” Above all, have fun and appreciate what we have been given and this opportunity to improve ourselves and open up other doors of understanding, which math does.
P1 = Part 1 Lesson P2 = Part 2 Lesson P3 = Part 2 Lesson
The number following the letter is the Lesson Number.
E = Exercises with Answers: Answers are in brackets [ ]. EA = Exercises Answers: (only used when answers are not on the
same page as the exercises.) Answers are in brackets [ ]. 1.2 A Message to the student regarding problems As you know, the more you practice the better and faster you will get, just like in any game or sport. So these problems are for those of you who want to get better and better. If you solve the problems given in the Simmons book you will be OK and be able to pass the Quiz, and probably do pretty good on the SAT and ACT. But, the more you practice the better you’ll get. And, I find I enjoy math more the better I get. I have included some problems that should also help you understand the math behind them better and better. Some of these problems may be pretty challenging, although you do have enough knowledge to solve them if you think long enough. DO NOT get discouraged when you tackle a hard problem and struggle with it. THAT HAPPENS to all mathematicians. In fact, we really seem to be understanding math at a deeper level when we struggle with a problem. If you get frustrated, take a break. If you have others to work with help each other. You will learn a lot by teaching and explaining things to others. That is really how I learned math. I started “tutoring” some of my classmates when I was 15 years old. And, they helped me too. Learning math is like any sport, the more we practice and coach others the better we get.
So, I hope you enjoy these additional problems. IF YOU FIND A MISTAKE in my answers, Bully for you. I’m pretty sure there will be some mistakes creeping in from time to time. Sometimes, I leave them in for you to find a little gem. It will help you build your confidence, too. Never believe something just because it is written in a textbook. Verify it with your own thinking and understanding as much as you can. I can’t tell you how many “mistakes” I have found in various textbooks over the years. And, sometimes on tests, too. You might want to practice my Motto: “Never be Satisfied, Always be Content.” Above all, have fun and appreciate what we have been given and this opportunity to improve ourselves and open up other doors of understanding, which math does.
P1-1 LESSON: The Real Number System P1-1E In Simmons’ book on p.36, do Exercise 1. Place each number in the smallest possible set N, Z, Q, R, C (Natural Numbers, Integers, Rational Numbers, Real Numbers, Complex Numbers) 1. 89
Place each number in the smallest possible set N, Z, Q, R, C (Natural Numbers, Integers, Rational Numbers, Real Numbers, Complex Numbers) 1. 89 [N Positive Integer]
P1-4 LESSON: Root, Radical, Fractional Exponents P1-4E In Simmons’ book on p.41-42, do Exercises 8-9, on p.43, do Exercises 10-11. Simplify or Calculate (Check with calculator):
P1-7 LESSON: Linear Equations & Rule of Algebra P1-7E In Simmons’ book on p.49, do Exercises 17-18. Write out the Rule of Algebra.
What is NOT permissible?
Solve for the obvious unknown or x. You might have to review Algebra from the Practical Math Foundation. It’s pretty amazing how much algebra you already have learned. Just apply the Rules of Algebra you already know, or should remember.
P1-7EA Linear Equations & Rule of Algebra Answers: [ ]
Write out the Rule of Algebra.
[If you have an equation LS = RS you may create a new equivalent equation by performing the same mathematical operation on BOTH sides
of the original equation.]
What is NOT permissible? [Division by 0]
Solve for the obvious unknown or x. You might have to review Algebra from the Practical Math Foundation. It’s pretty amazing how much algebra you already have learned. Just apply the Rules of Algebra you already know, or should remember.
1. What are the two roots or solutions of: Ax2 + Bx + C = 0
[ [-B+(B2–4AC)1/2]/2A ; [-B-(B2–4AC)1/2]/2A ]
2. What are the two roots or solutions of: Px2 + Qx + S = 0
[-Q+(Q2–4PS)1/2]/2P ; [-Q-(Q2–4PS)1/2]/2P]
3. What are the two roots or solutions of: Cx2 + Ax + B = 0
[-A+(A2–4CB)1/2]/2C ; [-A-(A2–4CB)1/2]/2C]
This is why formulas can be confusing. You must be sure what each of the parameters or constants really stand for. #1 is how the quadratic formula is usually displayed. But, these other two would be just as good, just different constants for the coefficients.
Solve for the solutions or roots of:
4. x2 + x - 2 = 0 [x = -2, x = 1 Could solve by factoring]
5. x2 + x - 3 = 0 [x = -1/2 + √13/2 or x = -1/2 - √13/2]
6. x2 + 2x - 3 = 0 [x = -3 or x = 1 Could solve by factoring]
7. x2 - 2x - 2 = 0 [x = 1 + √3 or x = 1 - √3]
8. 2x2 + 2x - 3 = 0 [x = -1/2 + √28/4 or x = -1/2 - √28/4] You might note: √28 = √4x√7 = 2√7
9. 2y2 + 2y - 3 = 0 [y = -1/2 + √28/4 or y = -1/2 - √28/4]
10. 3x2 + 2x - 4 = 0 [x = -1/3 + √52/6 or x = -1/3 + √52/6] You might note: √52 = √4x√13 = 2√13
11. 3x2 - 2x + 4 = 0 [No real solution]
12. x2 + x + 1 = 0 [No real solution. i = √-1] x = -1/2 + √3/2i, or x = -1/2 + √3/2i
We will learn all about these “complex numbers” in Tier 4. They are very important in STEM subjects. 13. x3 – 1 = 0 [x = 1, plus two solutions in #12] All of these solutions are called the 3rd Roots of Unity since they all satisfy x3 = 1
14. x5 – 1 = 0 [x = 1, and…] [ …the other four 5th roots of unity, which cannot be found with the quadratic formula. This problem is fairly easy to solve with trigonometry when it is extended to complex numbers in Tier 4. ]
15. 2x4 + 2x2 - 3 = 0
Hint: let y = x2 and solve for y first, then for x. 2y2+2y-3=0 See Problem # 9. Note the second root is negative and thus has no real square root. So we have two solutions now:
x = [-1/2+√28/4]1/2 and x = -[-1/2+√28/4]1/2 If there had been two positive roots, then there would have been four real number solutions. Example: (x-2)(x-3)=x2 – 5x + 6 = 0 has two roots, x =2 and x = 3. So x4 – 5 x2 + 6 = 0 will have four real roots: