Math 111!
Algebra! Prerequisite: None
Course Description
In this micro-course, students will review typical algebra concepts and problems. We will begin with simultaneous equations, functions, and multiple function problems, and review distance, median/mode, and absolute value.!
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Simultaneous Equations
Simultaneous Equations! ! Identify:! You will see more than one equation containing more than one variable (often x and y).! ! Set Up:! First, try to add or subtract the equations. If that’s not possible, solve for one of the variables and plug it into the other equation. ! ! Make Sure:! Look for easy substitutions and cancellations. ! ! Execute:! Answer the question they’re asking. Be careful: they may not be asking for x or y, but (x – y) or x 2.!
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Simultaneous Equations
Example:! 2x + 4y = 8 3x – 4y = 7 In the system of equations above, what is the value of 4y? (A) (B) (C) (D) (E)
2 4 6 8 10
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Function Problems
Function Problems – Tables! ! Identify:! Look for problems involving a table of variables (x) and functions f(x).! ! Set Up:! PLUG: Pick the simplest variable value from the table to plug into each of the answer choice functions. ! ! Make Sure:! Be careful of negatives and fractions when plugging into the answer choices.! ! Execute:! CHUG: Solve each answer choice using your value until you find the function that gives you the right result.!
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Function Problems
Example:! x
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f(x)
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–3
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The table above gives values of the linear function f(x) for selected values of x. Which of the following defines f(x) ? (A) (B) (C) (D) (E)
f(x) = –x + 3 f(x) = 3x – 4 f(x) = 2x + 3 f(x) = 2x – 3 f(x) = –x + 4
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Function Problems
Function Problems – Multiple Functions! ! Identify:! When they give you more than one function, and you have to combine them.! ! Set Up:! Find the FUNCTION, then find the QUANTITY being plugged in. ! ! Make Sure:! Be careful to solve the functions one at a time, beginning with the innermost parentheses.! ! Execute:! PLUG the QUANTITY into the FUNCTION. SOLVE by combining simultaneous equations.!
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Function Problems
Example:! 1 Let the function f be defined by f ( x) = x 2 + 4. If f (3p) = 8p, 3 what is one possible value of p?
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Absolute Value Problems
Absolute Value Problems! ! Identify:! Any problem that uses the | … | absolute value bars.! ! Set Up:! |x+3|=7 Create two equations:! ! ! ! !
x+3=7 x=4
x + 3 = –7 x = –7 + –3 x = –10
! Make Sure:! Be careful with negatives. If the question is an INEQUALITY, don’t forget to FLIP THE DIRECTION of the inequality sign for the negative case. ! ! Execute:! Solve the two equations for the variable.!
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Absolute Value Problems
Example:! b
a -3
-2
-1
c 0
d 1
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If a, b, c, and d are the coordinates of the indicated points on the number line above, which of the following is the smallest? (A) (B) (C) (D) (E)
|a+d| |d–a| |a+b| |c–b| |c+a|
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Statistics
Median and Mode Problems! ! Identify:! Look for words “median” and “mode”.! ! Set Up:! Make a list of all of the values in increasing order:! ! Make Sure:! Be sure that you have included all of the values in the set. ! ! Execute:! If you are finding the MEDIAN, pick the value in the middle of the list. If you are finding the MODE, pick the value(s) that appear most often.!
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Rate Problems
Rate Problems! ! Identify:! Look for problems involving distances and/or times.! ! Set Up:! List the elements you know, and plug them into the rate formula:! Distance = Rate x Time! ! Make Sure:! Keep your UNITS straight! Make sure you include all the elements.! ! Execute:! Just do the math.!
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Review Questions If Monica ran at an average rate of 8 miles per hour for 24 miles and Renee ran half as fast as Monica for the same distance, how much longer did it take Renee to run the distance than Monica? (A) 1 hour (B) 3 hours (C) 6 hours (D) 8 hours (E) 24 hours
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Review Questions If a – 7 = 2b and a = 5 + 3b, what is the value of b? (A) (B) (C) (D) (E)
12 5 2 –2 –5
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Review Questions In a certain student group, the median age of the members is 18. Which of the following must be true? I. The mode of the members’ ages is 18. II. The oldest member in the group is at least 1year older than the youngest member. III. If there is a 17-year-old in the group, there is also a 19year-old. (A) (B) (C) (D) (E)
None I only II only III only I and III
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Review Questions If | x – 4| = 6, which of the following could be a value of x? (A) (B) (C) (D) (E)
–2 2 4 8 12
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Review Questions If (a + 2)2 = 25 and a > 0, what is the value of a2 ?
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Review Questions If 2a2 – 5 = 67, what is the sum of all possible values of a?
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Review Questions For all numbers a and b, let the operation ¤ be defined by a ¤ b = 2ab – 4a. If a and b are positive integers, which of the following can be equal to zero? I. a ¤ b II. (a – b) ¤ b III. b ¤ (a – b) (A) (B) (C) (D) (E)
I only II only III only I and II only I, II, and III
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Review Questions If the average of a, 2a – 8, 2a + 2, 3a – 1, and 4a + 1 is 6, what is the value of the mode of these numbers?
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Review Questions For which of the following functions is f (–1) > f (1)? (A) (B) (C) (D) (E)
f (x) = 1 f (x) = 3(x – 2) f (x) = 7 – 3x f (x) = 5x2 f (x) = 2 – x2
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Review Questions Edward and Jacob work on an assembly line that produces widgets. Edward can make 24 widgets in one hour, and Jacob can make 16 widgets, every half-hour. If they work together for 7.5 hours, how many widgets can they make? (A) (B) (C) (D) (E)
120 180 210 240 420
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