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LEAP 2025 ALGEBRA I
BOOT CAMP
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Louisiana 1st Edition
Table of Contents Chapter 1: LEAP 2025 Algebra I Overview .................................................... 7
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Chapter 2: Solving Algebraically ......................................................................... 13 Solving Algebraically Overview ......................................................................... 14 Mini-Test One ............................................................................................................... 16 Plug it in .......................................................................................................................... 18 Distributive Property: Show Your Work .......................................................... 19 Mini-Test Two ............................................................................................................. 21 Create a Visual ........................................................................................................... 23 Substitution .................................................................................................................. 24 Negative Paranoia ..................................................................................................... 25 Mini-Test Three .......................................................................................................... 26 Mini-Test Four ........................................................................................................... 28 Try Numbers ................................................................................................................. 30 Mini-Test Five .............................................................................................................. 31 Word Problem Translation .................................................................................. 33 Mini-Test Explanations .......................................................................................... 34 Chapter 3: Interpreting Functions .....................................................................41 Functions Overview .................................................................................................. 42 Mini-Test One ............................................................................................................. 43 Get Real ........................................................................................................................ 45 Don’t Overthink It .................................................................................................... 46 Mini-Test Two ............................................................................................................. 47 Read the Question ..................................................................................................... 50 Domain ............................................................................................................................. 51 Mini-Test Three .......................................................................................................... 52
Use the Answer Choices......................................................................................... 54 Mini-Test Four ............................................................................................................ 55 Mini-Test Explanations ........................................................................................... 57
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Chapter 4: Solving Graphically and Rate of Change ............................... 61 Solving Graphically and Rate of Change Overview ................................62 Mini-Test One ............................................................................................................. 63 Finding Percentages ................................................................................................. 66 Equations of Lines ..................................................................................................... 68 Process of Elimination ............................................................................................ 70 Mini-Test Two .............................................................................................................. 71 Plug in Points on a Graph ..................................................................................... 73 Multi-Step Panic ........................................................................................................ 75 Mini-Test Three .......................................................................................................... 77 Translations & Reflections .................................................................................. 80 Mini-Test Explanations .......................................................................................... 82 Chapter 5: Wrap-Up ...................................................................................................87
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Chapter 6: Further Practice ..................................................................................91 Practice Set One .......................................................................................................92 Practice Set Two ....................................................................................................... 95 Practice Set Three .................................................................................................... 99 Practice Set Explanations .................................................................................. 103
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Fifteen years ago, I made perfect scores on the ACT and the SAT. Since then, I’ve helped thousands of students improve their scores on the most popular standardized tests, including the ACT, SAT, and WorkKeys. I’ve helped students who aren’t “good test takers” get the scores they need for the school, scholarship, or job they’re interested in.
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This Boot Camp was written to help you. If I were tutoring you personally, you might ask, “I’m taking my end-of-course test for Algebra I in two days. What do I need to know?” The information in this workshop is exactly what I would give you. This program is designed to give you the ultimate one-day cram. With this Boot Camp, you can learn the skills that you need to boost your scores on the Algebra I end-ofcourse test. Most students I work with do best under pressure. You can call it simply “waiting until the last minute,” but I think there is a human instinct that kicks in, a survival mechanism of sorts, that clears your head and helps you do what needs to get done. This workshop is for the “final hours.” It’s here because, sometimes, the best prep comes just before the test begins. This Boot Camp will guide you through the essentials on content, test-taking strategies, and the most common question types on the Algebra I end-of-course test. At MasteryPrep we’ve researched what students most need to do well on an end-of-course test and have distilled that information down into this workshop. It’s designed to give you everything you need— and nothing that you don’t—quickly enough that you can cover it all in a single day.
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Good luck!
Craig Gehring
LEAP 2025 Algebra I Boot Camp
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WORKKEYS OVERVIEW
A Note from the Author
Fill in the times following your instructor’s directions. This is the agenda we will follow throughout the day. There will be breaks throughout the session. Next to each section name in the schedule, you’ll find the corresponding page number where it begins in this workbook.
Section
Page Number
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Time
LEAP 2025 Algebra I Overview
7
Solving Algebraically
13
Break
—
Interpreting Functions
41
Break
—
Solving Graphically and Rate of Change
61
Boot Camp Wrap-Up
91
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WORKKEYS OVERVIEW
Schedule
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LEAP 2025 Algebra I Boot Camp
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Chapter 1
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LEAP 2025 Algebra I Overview
What Is End-of-Course Testing? End-of-Course (EOC) testing measures your aptitude in a given subject after you have finished a course. Consider it a subject understanding checkup. Teachers use it to identify both your strengths and areas where improvement is needed. This helps ensure you are on track in developing the knowledge and skills needed for the next grade and, eventually, college and a career.
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Why Should You Care?
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In this Boot Camp, we’ll focus on the three most common concepts in Algebra I: Solving Algebraically, Interpreting Functions, and Solving Graphically and Rate of Change. Your understanding of each of these concepts will help you pass the LEAP 2025 Algebra I test.
•
Many schools require the EOC as part of your final grade in the course.
•
If your school uses the EOC as a final exam for the course, then doing well on this test can boost your GPA.
•
A good EOC score is a positive indicator that you are on track for college.
•
Mastering the foundational skills taught in this Boot Camp will help you succeed in more difficult math courses in the future.
•
Put in the effort now and save yourself from repeating a course or taking summer school.
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LEAP 2025 ALGEBRA I OVERVIEW
LEAP 2025 Algebra I Overview
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LEAP 2025 Algebra I Boot Camp
Orientation
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The three most common conceptual categories tested on LEAP 2025 Algebra I are Solving Algebraically, Interpreting Functions, and Solving Graphically and Rate of Change. The test is timed. All sections except the first one allow you to use a calculator. To do well on this test, it is important to be comfortable performing mathematical operations with and without a calculator. Here is a breakdown of possible ways the LEAP 2025 Algebra I assessment will test you in each of the three main categories. Each section of the test has questions from each category.
Solving Algebraically tests equations, expressions, and inequalities in the following ways: Evaluate problems with one or two variables
•
Create models to describe real-life situations and relationships
•
Understand and apply basic mathematical principles
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•
Interpreting Functions measures your ability to interpret, understand, and build functions. Solving Graphically and Rate of Change assesses how well you can create graphs and solve problems involving lines, slope, intercepts, and solution sets.
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LEAP 2025 ALGEBRA OVERVIEW
LEAP 2025 Algebra I Overview
Orientation In Louisiana, your LEAP 2025 Algebra I score can count as a percentage of your final grade for the Algebra I course. The percentage is always between 15 and 30 percent of your grade, depending on your school district. If you know what areas you struggle in, compare them to the most important skills needed for the test. The table below gives the approximate contribution of each conceptual category to your score. This is just an estimate and can vary greatly from one test to another. Total Points
Percentage of Points
Solving Algebraically
27
40%
Interpreting Functions
17
Solving Graphically and Rate of Change
24
35%
Total
68
100%
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Conceptual Category
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25%
The test is administered in four sessions. The sessions are timed, so it is important to manage your minutes and avoid spending too much time on any one question. Usually, each single-part question is worth one point, and each multi-part question is worth one point per part. Questions that require you to show work or justify your answers are typically worth 2, 3, or even 4 points each. Each test section may include multiple choice, multiple select, constructed response, fill in the blank, or a variety of technology-assisted answering methods. For example, you might have to actually draw a graph on your computer. We provide practice for all of these different question types during the Boot Camp.
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LEAP 2025 ALGEBRA I OVERVIEW
LEAP 2025 Algebra I Overview
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Test Session
Calculator?
Number of Points
Time Limit
Session 1a
No
9
25 minutes
Session 1b
Yes
13
55 minutes
Session 2
Yes
23
80 minutes
Session 3
Yes
23
80 minutes
Total
–
68
240 minutes
LEAP 2025 Algebra I Boot Camp
About This Boot Camp This is a one-day event preparing you for the most important questions on the LEAP 2025 Algebra I test. This Boot Camp is not meant to be your only form of preparation. You should also find other practice tests online and ask your teachers for help and other resources that specifically target the skills needed to do well on this test.
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This book contains key strategies for taking the test, instructional content, and mini-tests that give you practice with the types of questions you’ll see on test day (plus an explanation for how to solve each problem). This book has plenty of places for you to take notes, and we highlight the most important strategies to use so you can continue practicing on your own.
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This Boot Camp will go by fast! Be ready, take notes, and stay focused!
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LEAP 2025 ALGEBRA OVERVIEW
LEAP 2025 Algebra I Overview
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Chapter 2
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Solving Algebraically
Solving Algebraically:
Overview
Solving Algebraically Overview The Solving Algebraically conceptual category tests your proficiency over a broad range of algebra skills. The
Interpret the structure of expressions
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Write expressions in equivalent forms to solve problems
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Seeing Structure in Expressions
Arithmetic with Polynomials and Rational Expressions •
Perform arithmetic operations on polynomials
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Understand the relationship between zeros and factors of polynomials
Creating Equations •
Create equations that describe numbers or relationships
Reasoning with Equations and Inequalities •
Understand solving equations as a process of reasoning and explain the reasoning
•
Solve equations and inequalities in one variable
•
Solve systems of equations and inequalities
•
Represent and solve equations and inequalities graphically
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SOLVING ALGEBRAICALLY
skills that will be tested on your exam include but are not limited to the following:
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LEAP 2025 Algebra I Boot Camp
Solving Algebraically:
Overview
What Are Boot Camp Mini-Tests? During this Boot Camp you will take several mini-tests, which are small segments of an Algebra I test. While taking these mini-tests, it’s important to imagine that you are in an actual testing environment. The time limits assigned as you complete the mini-tests. In the mini-tests, we are focusing on only one category of questions at a time, but on the real assessment each test section will include questions from each major conceptual category.
For these mini-tests, you have 10 minutes to answer several questions. Your instructor will signal when you are out of time. Try to get through all the questions within the time limit. Unless your instructor has provided you with
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an answer sheet, circle your answers directly in this book. The real test does not allow the use of cell phones, watches, or computers, so you shouldn’t use them on the mini-tests, either.
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SOLVING ALGEBRAICALLY
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match the pace that you should try to keep during the actual test. Practice all of the skills that you have learned
Solving Algebraically - Mini-Test One 1.
Consider the function f, where f(t) = 2t2 + 8t – 10.
PART A What is the vertex form of f(t) ? A. 2(t – 2)2 – 18 B. 2(t + 2)2 – 18 C. 2(t – 4)2 – 14
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D. 2(t + 4)2 – 14
PART B What is a factored form of f(t) ?
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A. (2t – 1)(t + 10) B. (2t + 1)(t – 10) C. 2(t + 5)(t – 1)
D. 2(t – 5)(t + 1)
2.
Which expression is equivalent to (x – 2)(3x 2 – 5x + 9)? A. 3x 2 – 4x + 7 B. 3x 2 – 4x + 9
C. 3x 3 – 5x 2 + 9x – 2
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D. 3x 3 – 11x 2 + 19x – 18
Solving Algebraically - Mini-Test One
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4.
The boiling point of ethyl alcohol, T (measured in degrees), at an altitude above sea level, a (measured in feet), can be determined by the expression –0.0013a + 173. What is the meaning of the 173 in the expression? A.
The boiling point is 173 degrees at sea level.
B.
The boiling point decreases by 173 degrees as the altitude increases by 1,000 feet.
C.
The minimum altitude is 173 feet.
D.
The maximum altitude is 173 feet.
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3.
Kedrick used the method of completing the square to solve a quadratic equation. His first two steps are shown below.
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Given: 3x2 + 30x + 21 = 0 Step 1: x2 + 10x + 7 = 0 Step 2: x2 + 10x = –7
Write numbers in each box to correctly complete the square in Step 3 (selecting from the options appearing below). Step 3: x2 + 10x +
–32
–25
–18
–5
5
18
25
32
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Options:
=
Solving Algebraically - Mini-Test One
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STOP! END OF TEST. YOU MAY GO BACK AND CHECK YOUR WORK.
Solving Algebraically:
Plug It In
Plug It In If you don’t know exactly how to answer a question, try plugging in assumed values for the variables given and see if you can draw some conclusions.
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The boiling point is 173 degrees at sea level.
B.
The boiling point decreases by 173 degrees as the altitude increases by 1,000 feet.
C.
The minimum altitude is 173 feet.
D.
The maximum altitude is 173 feet.
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In this question, plug in experimental values for a to prove answer choices incorrect. Choice B says that an increase of 1000 in a will increase the value of the expression by 173. Test to see if this is true. Use 1000 and 2000 as values for a and see what happens.
–0.0013(1000) + 173 = 171.7 and –0.0013(2000) + 173 = 170.4. This is not a difference of 173, so we can eliminate choice B.
Choices C and D can be eliminated because you can plug in a value of 150 (below the “minimum altitude” described in C) and a value of 200 (above the “maximum altitude” described in D) and the expression still works. Therefore, choice A is correct. You can verify this by plugging in the value 0 (sea level) for a. This gives you a boiling point of –0.0013(0) + 173 = 173, which is exactly what choice A claims.
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SOLVING ALGEBRAICALLY
3. The boiling point of ethyl alcohol, T (measured in degrees), at an altitude above sea level, a (measured in feet), can be determined by the expression –0.0013a + 173. What is the meaning of the 173 in the expression?
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LEAP 2025 Algebra I Boot Camp
Solving Algebraically:
Distributive Property: Show Your Work
Distributive Property: Show Your Work Use the distributive property when you need to get rid of parentheses.
With the distributive property, you actually distribute (share) the coefficient outside of the terms in parentheses among each of the individual elements inside the parentheses. Share the wealth!
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For example: 3(2x + 5) = (3)(2x) + (3)(5) (3)(2x) + (3)(5) = 6x + 15
Be cautious about keeping track of the negative signs. –4(x – 4) = (–4)(x) – (–4)(4)
(–4)(x) – (–4)(4) = –4x – (–16) –4x – (–16) = –4x + 16
It is important to show your work throughout the entire process of solving these problems. If you try to rush and
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skip steps, you increase your chance of making a careless error.
Let’s take a look at this tricky question:
2.
Which expression is equivalent to (x – 2)(3x 2 – 5x + 9)? A. 3x 2 – 4x + 7 B. 3x 2 – 4x + 9
C. 3x 3 – 5x 2 + 9x – 2
D. 3x 3 – 11x 2 + 19x – 18
LEAP 2025 Algebra I Boot Camp
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SOLVING ALGEBRAICALLY
parentheses, this is a clue that you’ll need to use this method.
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When you are asked to simplify an expression that has parentheses and none of the answer choices have
Solving Algebraically:
Distributive Property: Show Your Work
(x – 2)(3x2 – 5x + 9) is equivalent to x(3x2 – 5x + 9) – 2(3x2 – 5x + 9). First, distribute the x to each term in the second polynomial. Then distribute the –2 to every term in the second polynomial. 3x3 – 5x2 + 9x – 6x2 + 10x – 18 Combine like terms.
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By working one step at a time, you can avoid the mistakes that cause inaccuracies on your exam.
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3x3 – 11x2 + 19x – 18
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3x3 – 5x2 – 6x2 + 9x + 10x – 18
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LEAP 2025 Algebra I Boot Camp
Solving Algebraically - Mini-Test Two 1.
b1 + b2 h, where A is the area, b 1 and b 2 are the lengths of the 2
The formula for the area of a trapezoid is A = bases, and h is the height. Which is the formula for b 2?
b1
b2
B. C. D.
2.
b2= b1 −
2A h
= b2
2A − b1 h
b2 =
b1 + A h 2
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A.
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h
b = 2b1 + 2 Ah 2
What is the solution to –3(6t + 4) – 6t ≥ –11t – (17t + 4)? A. t ≤ 2 B. C.
t≤4 t≥4
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D.
t≥2
Solving Algebraically - Mini-Test Two
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3.
Rayan and Megan are playing a game. •
Rayan and Megan each started with 50 points.
•
At the end of each turn, Rayan’s points increased by 250.
•
At the end of each turn, Megan’s points doubled.
PART A Create a model that can be used to determine the total number of points between Rayan and Megan based on the number of turns that have passed.
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Write your model in the space below.
PART B
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At the end of the game, Megan has 400 points. How many points did Rayan and Megan score in total? Provide your answer in the space below. Show your work.
PART C
At the end of which turn does Megan’s score first exceed Rayan’s score?
Solving Algebraically - Mini-Test Two
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STOP! END OF TEST. YOU MAY GO BACK AND CHECK YOUR WORK.
Solving Algebraically:
Create a Visual
Create a Visual It can be difficult to visualize word problems on a math test. Many problems that involve a picture or shape don’t actually show the picture in your test booklet. If the path to solving a question doesn’t immediately pop out at you, creating a visual of the information in the question can make it more obvious.
Rayan and Megan are playing a game. Rayan and Megan each started with 50 points.
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• •
At the end of each turn, Rayan’s points increased by 250.
•
At the end of each turn, Megan’s points doubled.
PART C: At the end of which turn does Megan’s score first exceed Rayan’s score?
Use the information given in the question to draw a table relating the turns to the number of points. Both players start out with 50 points. Megan’s points double each turn, and Rayan’s points increase by 250 each turn. Write out the number of points both people have for each turn until Megan’s points exceed Rayan’s.
Megan
Turn 1
Turn 2
Turn 3
Turn 4
Turn 5
50
100
200
400
800
1,600
50
300
550
800
1,050
1,300
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Rayan
Start
Because you created a visual, you can determine that Megan has more points than Rayan at the end of the 5th turn. You didn’t have to use any algebra at all!
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SOLVING ALGEBRAICALLY
3.
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Let’s take a look at a problem that dramatically decreases in difficulty once you create a visual.
Solving Algebraically:
Substitution
Substitution Freezing up on a challenging question is a common problem among students. When you come across a scary-looking question and don’t know where to begin, look for a formula. If you find a formula, see if there are any values to substitute into it. Use this as a starting point for figuring out the question. If the problem doesn’t give any values, you can assume (make up) values!
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provided.
The formula for the area of a trapezoid is A = of the bases, and h is the height.
b1 + b2 h, where A is the area, b 1 and b 2 are the lengths 2
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1.
Which is the formula for b 2?
b1
h
b2
2A h
A.
b2= b1 −
B.
= b2
2A − b1 h
C.
b2 =
b1 + A h 2
D.
b = 2b1 + 2 Ah 2
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For example, consider how this problem becomes manageable if you substitute made-up values into the formula
8+ 4 · 2 = 12. You now have a full set of values for the variables. 2 If an answer choice doesn't work for those values, it's wrong. Plug these values into each answer choice and Assume that b1 is 8, b2 is 4, and h is 2. Then A =
eliminate the ones that don't work.
Choice A can be eliminated because it gives the value for b2 as –4 instead of 4. b2 = 8 –
Choice C can be eliminated because with the assumed values it gives b2 =
2(12) = –4 2
8 + 12 · 2 = 20. 2
Choice D can be eliminated because it gives b2 = 2(8) + 2(12)(2) = 64 with the assumed values. Only choice B checks out with the assumed values, so it is correct. b2 = 2(12) – 8 = 4 2 Many math questions can be answered using this strategy. Assume values, and then see what choices work.
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LEAP 2025 Algebra I Boot Camp
Solving Algebraically:
Negative Paranoia
Negative Paranoia When solving problems loaded with negative signs, let the paranoia sink in. This test uses a lot of negative signs to trip you up. When you face a problem like this, you need to pay more attention to every calculation that involves a negative sign. It is likely that the trap answers are meant to catch you if you mess up a calculation with a negative sign.
What is the solution to –3(6t + 4) – 6t ≥ –11t – (17t + 4)?
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A. t ≤ 2 B.
t≥2
C.
t≤4
D.
t≥4
Simplify both sides of the inequality and solve for t. Don’t forget to be paranoid about the negative signs! –3(6t + 4) – 6t ≥ –11t – (17t + 4) –18t – 12 – 6t ≥ –11t – 17t – 4 –24t – 12 ≥ –28t – 4 28t – 24t ≥ 12 – 4
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4t ≥ 8 t≥2
Notice that if you forget to distribute the negative sign to the 4 at the end of the inequality, you will ultimately arrive at choice D, t ≥ 4. Don’t fall into these traps! Keep a close eye on all negative signs.
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SOLVING ALGEBRAICALLY
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Let’s look at an example of when you should feel the negative paranoia:
Solving Algebraically - Mini-Test Three 1.
Tasha deposited m money into a savings account y years ago. Now she is going to use some of the money in the savings account to buy a laptop. The following expression can be used to determine the percentage of the money in savings that Tasha will use for the laptop. 299 m (1.25 )
y
× 100
What is the meaning of the denominator in the expression? The amount in Tasha’s savings account after the laptop purchase
B.
The yearly interest rate for the savings account
C.
The amount Tasha will pay for the laptop
D.
The amount in Tasha’s savings account now
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A.
2. Let x represent an irrational number and let y represent a non-zero rational number. PART A
Which expression could represent a rational number? A. B.
x–y xy 2
C. –x D.
x2
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PART B
A quadratic equation with integer coefficients has two distinct zeros. If one zero is rational, which statement is true about the other zero? A.
The other zero must be irrational.
B.
The other zero must be rational.
C.
The other zero must be non-real.
D.
The other zero can be either rational or irrational.
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Gerald received a 3% annual increase in his pension fund for each of 5 years. He plans to continue working at the same job for an additional x years. If he continues to earn a 3% annual increase in his pension fund, which statement gives the expression that can be used to calculate the total percent increase in his pension from the first year to the last year? A.
The expression 1.03(5 + x) can be used because 1.035 • 1.03 x = 1.03(5 + x)
B.
The expression 1.03(5x) can be used because (1.035 )x = 1.03(5x)
C.
The expression 1.03(5x) can be used because 1.035 • 1.03x = 1.03(5x)
D.
The expression 1.03(5 + x) can be used because 1.035 + 1.03 x = 1.03(5 + x)
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3.
Two real numbers are defined below.
s = 0.707106781186 ... v = 0.222222222222 ...
Determine whether each number is rational or irrational. Is the quotient of s and v rational or irrational?
Justify your answers.
Write your answers and your justifications in the space provided below.
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Solving Algebraically - Mini-Test Three
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STOP! END OF TEST. YOU MAY GO BACK AND CHECK YOUR WORK.
Solving Algebraically - Mini-Test Four 1.
A grocery store purchases crates of oranges. •
Each crate contains 75 oranges.
•
Each crate costs $60.
How much does the grocery store have to charge for each orange to make a profit of $30 per crate? A. $0.75 B. $0.83 C. $0.90
Select the box that correctly identifies the best description of the solution or solutions for each equation.
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2.
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D. $1.20
x = –3
x=1
No solution
Infinite solutions
3(3x + 5) = 4x + 5x – 8
5(x – 5) + 17 =
1 (10x – 6) – 5 2
–11x – 24 = –11x + 8(–2 – x)
1 1 (–10x – 5) = – (9x + 18) 5 3
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–2 +
3.
What is the largest of 3 consecutive positive integers if the product of the two smaller integers is 8 more than 2 times the largest integer?
Solving Algebraically - Mini-Test Four
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A.
–36 meters
B.
–2 meters
C.
6 meters
D.
18 meters
Kaitlyn organizes performances for a small theater company. She knows if she charges $25 per ticket, around 130 people will buy tickets. For every $1.00 she lowers the ticket price, an additional 5 people will purchase tickets. Kaitlyn wrote an expression showing the total income from ticket sales, where d is the number of dollars by which the ticket price has been reduced.
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5.
The height of a bird after s seconds is given by the expression 6 + 3(s – 2) 2. In the simplified form of the expression, the constant represents the initial height, in meters, of the bird. What is the initial height of the bird?
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4.
(25 – d)(130 + 5d)
Which of the following expressions is equivalent to Kaitlyn’s? A. 5d 2 – 130d + 3,250 B. –5d 2 – 5d + 3,250 C. –5d 2 + 3,125
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D. 4d + 155
Solving Algebraically - Mini-Test Four
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STOP! END OF TEST. YOU MAY GO BACK AND CHECK YOUR WORK.
Solving Algebraically:
Try Numbers
Try Numbers Sometimes you have to try numbers that aren’t given in the question or answer choices. In such a situation, you will have to read the question and determine which number should be used.
The height of a bird after s seconds is given by the expression 6 + 3(s – 2) 2. In the simplified form of the expression, the constant represents the initial height, in meters, of the bird. What is the initial height of the bird? –36 meters
B.
–2 meters
C.
6 meters
D.
18 meters
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A.
Instead of squaring the binomial (s – 2), you can plug in a number and solve. The question asks for the initial height of the bird, which would be found after 0 seconds have passed. Therefore, plug in 0 for s and simplify the expression.
6 + 3(s – 2)2
6 + 3(0 – 2)2 6 + 3(–2)2
6 + 3(4) = 6 + 12 = 18 meters
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SOLVING ALGEBRAICALLY
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Let’s take a look at a commonly missed word problem and see how we can unlock its answer:
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LEAP 2025 Algebra I Boot Camp
Solving Algebraically - Mini-Test Five 1.
What is the solution to –3(12z + 1) – 8z < –30z – (10z + 11)? 7 2 7 B. z > – 2
A. z < –
C. z < 2 z>2
2.
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D.
The expression (x – 2)(4x 2 – 5x + 7) is equivalent to which of the following? A. 4x 3 – 13x 2 + 17x – 14 B. C. D.
4x 2 – 4x + 7 4x 2 – 4x + 5
A runner spends w weeks training for a marathon. Her initial weekly training distance is 9 miles. Each week she adds 2.5 miles to her training distance. Which equation models the weekly training distance in miles (m) over time?
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3.
4x 3 – 5x 2 – 3x – 2
A. w = 2.5m + 9m B.
m = 2.5w + 9
C. w = 2.5m – 9
D. m = 2.5w – 9
Solving Algebraically - Mini-Test Five
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4.
Margot, the manager of a coffee shop, is planning to order aprons for each of her employees. There are 25 employees at the shop. The baristas request aprons with pockets. Margot determines that $200 is the maximum amount that can be spent on all of the aprons. Margot finds the following prices for aprons at a restaurant supply store. Option
Features
Cost per Apron
A
2 pockets
$6.50
B
4 pockets
$8.99
E
Using a to represent the number of aprons with 2 pockets and b to represent the number of aprons with 4 pockets, which of the following inequalities represents the constraint on the number of aprons that can be purchased based on the amount of money available to spend on aprons? A. 6.50a + 8.99b ≤ 200
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B. 6.50a + 8.99b ≥ 200 C. 2a + 4b > 200 D. 2a + 4b < 200
5.
Seven times Hector’s age minus two times Sandra’s age equals 5. Sandra’s age is also three times Hector’s
SA
age. How old is Sandra?
Solving Algebraically - Mini-Test Five
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Solving Algebraically:
Word Problem Translation
Word Problem Translation The secret to solving a word problem is translating it into math.
is, equal to, is the same as → =
times, product, each, per, of → •
minus, without, less, difference, change → –
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plus, together, and, combined, both → +
divided into, split between or among, divvied up → ÷
Let’s take a look at how word problem translation can help you solve a problem on your exam.
5.
Seven times Hector’s age minus two times Sandra’s age equals 5. Sandra’s age is also three times Hector’s age. How old is Sandra?
Translate the sentences into algebraic equations. Let variables represent the unknown ages and use the translations above to determine what math operations to use.
SA
7h – 2s = 5 s = 3h
Substitute 3h from the second equation for s in the first equation. 7h – 2(3h) = 5 7h – 6h = 5 h=5
Hector is 5 years old. Substitute 5 for h in the second equation. s = 3(5) = 15 Sandra is 15 years old. Remember to translate words into math! This approach can make even the hardest questions much easier to solve. LEAP 2025 Algebra I Boot Camp
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SOLVING ALGEBRAICALLY
operations.
E
When translating word problems to algebraic equations, it is important to know which words translate to which
Solving Algebraically:
Mini-Test Explanations
Mini-Test Explanations MINI-TEST ONE
Given a quadratic ax ² + bx + c, its vertex is found by computing h = –
h =
2t2 + 8t – 10 = 2(t + 2)2 + k
2t2 + 8t – 10 = 2(t2 + 4t + 4) + k
2t2 + 8t – 10 = 2t2 + 8t + 8 + k
−2
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−8 = 2(2)
b and then evaluating f(x) at h to find k. 2a
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k = –18
Thus the vertex form of this function is 2(t + 2)2 – 18. It’s t + 2 instead of t – 2 because h = –2 and the vertex form is given with x – h, not x + h.
Part B: The correct answer is C. Since all three terms are multiples of 2, first factor out 2. Then factor the remaining quadratic.
2(t2 + 4t – 5)
The binomial constants have a product of –5 (the last term in the quadratic) and a sum of 4 (the coefficient of the middle term in the quadratic). Their values must therefore be 5 and –1.
2(t + 5)(t – 1)
2. The correct answer is D. Distribute both terms in the first parentheses to all terms in the second parentheses and combine like terms.
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SOLVING ALGEBRAICALLY
1. Part A: The correct answer is B. The vertex form of a quadratic is given by f(x) = a(x – h)² + k, where (h, k) is the vertex.
(x – 2)(3x2 – 5x + 9)
3x3 – 5x2 + 9x – 6x2 + 10x – 18
3x3 – 5x2 – 6x2 + 9x + 10x – 18
3x3 – 11x2 + 19x – 18
3. The correct answer is A. Sea level is an altitude of 0. When a = 0, then the expression for the boiling point is –0.0013(0) + 173 = 173. Therefore, 173 represents the boiling point at sea level. 2
−b 4. The correct answers are 25 and 18. To complete Step 3, the term is added to both sides of the 2a equation, where a is the coefficient of the x2 term and b is the coefficient of the x term. In the quadratic equation provided, a = 1 and b = 10.
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LEAP 2025 Algebra I Boot Camp