Title: Midterm Exam
Course: Technical Math for Industry

Q: Which of the following will isolate the variable in the expression  ?
- Multiply by 6, then add 8. (Isolating the variable requires performing the inverse of each operation in the correct order. The inverse of +8 is -8, so you need to subtract 8. This results in 6x. The inverse of multiplying by 6 is dividing by 6, which gives you x by itself.)
- Divide by 6, then subtract 8. (You flipped the order in which the subtraction and division should take place. In this case, you would need to subtract 8, and then divide by 6 to isolate the variable.)
- Subtract 8, then divide by 6. (By subtracting 8 first, you are performing the inverse of +8, resulting in the expression 6x. By dividing by 6, you are performing the inverse of multiplying by 6, and completely isolating the variable x.)
- Add 8, then multiply by 6. (Isolating the variable requires performing the inverse of each operation in the correct order. The inverse of +8 is -8, so you need to subtract 8. This results in 6x. The inverse of multiplying by 6 is dividing by 6, which gives you x by itself.)
A: Subtract 8, then divide by 6.

Q: Which of the following methods will solve the equation   ?
- Add 6 to both sides of the equation and multiply both sides by . (Incorrect. Isolating the variable requires performing the inverse of the operations in the correct order.)
- Subtract 2 from both sides and then multiply both sides by 3. (Correct. You can isolate a variable by using inverse operations to first remove other terms in the expression and then to clear the coefficient of the variable. As long as the same operations are done to both sides of the equation, equality is maintained.)
- Subtract 6 from the left side of the equation and multiply the right side by 3. (Incorrect. The equality of an expression is lost if the same operations are not done to both sides of the equation.)
- Multiply both sides of the equation by 3 and then subtract 2 from both sides. (Incorrect. Multiplying both sides by 3 will result in . Subtracting 2 creates , so the variable is not isolated.)
A: Subtract 2 from both sides and then multiply both sides by 3.

Q: In order to clear away the fractions from , which of the following numbers can we use to multiply both sides of the equation?
- 10 (Incorrect. In this case, you must clear the fraction by multiplying them by a common denominator. 10 is not a common denominator of 7 and 3.)
- 62 (Incorrect. In this case, you must clear the fraction by multiplying them by a common denominator. 62 is not a common denominator of 7 and 3.)
- 15 (Incorrect. In this case, you must clear the fraction by multiplying them by a common denominator. 15 is not a common denominator of 7 and 3.)
- 21 (Correct. 21 is the lowest common denominator of 7 and 3. You can multiply both sides of the equation by this number to clear away the fractions.)
A: 21

Q: Which of the following is the correct value of ?
- 75 (Incorrect. The absolute value of a number is always positive or zero. If the original number is negative, its absolute value is that number without the negative sign. In this case, .)
- -15 (Incorrect. The absolute value of a number is always positive or zero. If the original number is negative, its absolute value is that number without the negative sign. In this case, .)
- 5 (Correct. .)
- -5 (Incorrect. The absolute value of a number is always positive or zero. If the original number is negative, its absolute value is that number without the negative sign. In this case, .)
A: 5

Q: What point on the number line is
- 30 (Correct. The absolute value of)
- 40 (Incorrect. The expression . What is the absolute value of -10? What do you get when you multiply it by 3?)
- 15 (Incorrect. The expression . What is the absolute value of -10? What do you get when you multiply it by 3?)
- 5 (Incorrect. The expression . What is the absolute value of -10? What do you get when you multiply it by 3?)
A: 30

Q: Which values for w make  a true equation?
- -4 and 4 (Incorrect. . Which values for w can make ?)
- 0 (Incorrect. . Which values for w can make ?)
- -1 and 1 (Correct. Those values for w make the equation true.)
- -5 and 5 (Incorrect. . Which values for w can make ?)
A: -1 and 1

Q: Which of the following number lines show the correct solution to the equation ?
A: (a choice with no text)
If wrong: Incorrect. Begin by multiplying both sides of the equation by -5, and then solve the equation for k.
If wrong: Incorrect. Begin by multiplying both sides of the equation by -5, and then solve the equation for k.
If right: Correct. When k = 5 or -5, the equation is true.
If wrong: Incorrect. Begin by multiplying both sides of the equation by -5, and then solve the equation for k.

Q: Multiply using the distributive property.
A: (a choice with no text)
If wrong: Incorrect. Remember that each term within the parentheses must be multiplied by the expression on the outside of the parentheses. You can think of the expression as .
If wrong: Incorrect. Remember that each term within the parentheses must be multiplied by the expression on the outside of the parentheses. You can think of the expression as .
If wrong: Incorrect. Remember that each term within the parentheses must be multiplied by the expression on the outside of the parentheses. You can think of the expression as .
If right: Correct. All of the terms within the parentheses have been multiplied by the term on the outside of the parentheses.

Q: Which of the equations listed below is equivalent to:
A: (a choice with no text)
If wrong: Incorrect. When applying the Distributive Property, all of the terms inside the parentheses must be multiplied by the value outside. Here, you multiplied both and by 9; try multiplying both terms by 18.
If wrong: Incorrect. When applying the Distributive Property, multiplication occurs with only the terms inside the parenthesis, not the other parts of the equation. Here, you multiplied but not or .
If wrong: Incorrect. When applying the Distributive Property, multiplication occurs with only the terms inside the parenthesis, not the other parts of the equation.
If right: Correct. All of the terms inside the parentheses have been multiplied by the correct value outside.

Q: Jordan's sporting goods store had 60 basketballs. Before selling any, they added an unknown number of basketballs to their stock, resulting in 155 basketballs total.

Which of the following equations best represents this situation?
A: (a choice with no text)
If wrong: Incorrect. Remember that unknown refers to the variable, and added implies addition, and total indicates the total number of basketballs. If they began with 60, how many did they add to get to 155?
If right: Correct. You put the variable and the two numbers in the proper positions.
If wrong: Incorrect. Remember that unknown refers to the variable, and added implies addition, and total indicates the total number of basketballs. If they began with 60, how many did they add to get to 155?
If wrong: Incorrect. Remember that unknown refers to the variable, and added implies addition, and total indicates the total number of basketballs. If they began with 60, how many did they add to get to 155?

Q: Two opponents playing chess start out with 32 pieces on the chess board. Each person loses an equal amount of pieces over 15 minutes. By this time, only 16 pieces remain. How many pieces did each person lose?

Translate this word problem into an equation, using x to represent the number of pieces that each person lost.
A: (a choice with no text)
If wrong: Incorrect. Remember that the game started with 32 pieces and each person lost x amount of pieces. The number of pieces lost is ; what number do you think this expression should be set equal to?
If right: Correct. The variable, coefficient, and constants are all in the proper positions.
If wrong: Incorrect. Remember that the game started with 32 pieces and each person lost x amount of pieces. The number of pieces lost is ; what number do you think this expression should be set equal to?
If wrong: Incorrect. Remember that the game started with 32 pieces and each person lost x amount of pieces. The number of pieces lost is ; what number do you think this expression should be set equal to?

Q: The formula for the perimeter (p) of an isosceles triangle is , where a represents the length of one side, and b represents the length of the base.

Write the formula in terms of b, the base.
A: (a choice with no text)
If wrong: Incorrect. Imagine you have an isosceles triangle with a perimeter of 33 and a side length of 12. How long is the base? How do you know?
If right: Correct. To find the length of b, you can subtract 2a from both sides, leaving .
If wrong: Incorrect. Imagine you have an isosceles triangle with a perimeter of 33 and a side length of 12. How long is the base? How do you know?
If wrong: Incorrect. Imagine you have an isosceles triangle with a perimeter of 33 and a side length of 12. How long is the base? How do you know?

Q: To form 3 adjacent squares (as in the figure), 10 toothpicks are required. How many toothpicks are required to form 50 adjacent squares?
- 301 (Incorrect. Try a smaller number of squares first (4 and 5, for example). Do you see in a pattern in the number of toothpicks required? How many "new" toothpicks are required when you add an additional square? How many additional squares need to be added to the given figure in order to get 50 squares?)
- 57 (Incorrect. Try a smaller number of squares first (4 and 5, for example). Do you see in a pattern in the number of toothpicks required? How many "new" toothpicks are required when you add an additional square? How many additional squares need to be added to the given figure in order to get 50 squares?)
- 200 (Incorrect. Try a smaller number of squares first (4 and 5, for example). Do you see in a pattern in the number of toothpicks required? How many "new" toothpicks are required when you add an additional square? How many additional squares need to be added to the given figure in order to get 50 squares?)
- 151 (Correct. Notice that each new square requires 3 more toothpicks (2 horizontal and 1 vertical). Since we need 47 more squares (starting at the given figure), 50 squares will require , which equals 151 toothpicks.)
A: 151

Q: Use inductive reasoning to find the 12th term in the sequence 5, 8, 11, 14, ...
- 19 (Incorrect. What is the difference between the first two numbers? The next two? If this pattern continues, how can you get from the 4th term to the 12th term? How many times do you need to add 3?)
- 38 (Correct. Since each term is 3 more than the one before it, to determine the 12th term, you can start with the 4th term, which is 14, then add 3 a total of eight times.)
- 36 (Incorrect. What is the difference between the first two numbers? The next two? If this pattern continues, how can you get from the 4th term to the 12th term? How many times do you need to add 3?)
- 34 (Incorrect. What is the difference between the first two numbers? The next two? If this pattern continues, how can you get from the 4th term to the 12th term? How many times do you need to add 3?)
A: 38

Q: A common mathematical pattern starts as 1, 1, 2, 3, 5, ... The first two terms are both equal to 1 and after that, each number in the sequence is the sum of the two numbers immediately before it. That is, the third term in the sequence is 2 (since that equals 1 + 1), the fourth term is 3 (since that equals 1 + 2), and the fifth term must be 2 + 3, which equals 5. Therefore, the first five terms in the sequence (called the "Fibonacci Sequence") are 1, 1, 2, 3, and 5.  What is the 10th term of the sequence?
- 21 (Incorrect. Recall that the first five terms of the sequence are 1, 1, 2, 3, and 5, so the 6th term must equal 3 + 5, which is 8, and the 7th term is 5 + 8, which equals 13. Now we know the first 7 terms are 1, 1, 2, 3, 5, 8, 13. Can you continue the process to find the 10th term of the sequence?)
- 55 (Correct. Knowing that the first 5 terms of the sequence are 1, 1, 2, 3, and 5, the 6th term must equal 3 + 5 = 8, the 7th term is 5 + 8 = 13, the 8th term is 8 + 13 = 21, the 9th term is 13 + 21 = 34, and the 10th term is 21 + 34 = 55.)
- 34 (Incorrect. Recall that the first five terms of the sequence are 1, 1, 2, 3, and 5, so the 6th term must equal 3 + 5, which is 8, and the 7th term is 5 + 8, which equals 13. Now we know the first 7 terms are 1, 1, 2, 3, 5, 8, 13. Can you continue the process to find the 10th term of the sequence?)
- 89 (Incorrect. Recall that the first five terms of the sequence are 1, 1, 2, 3, and 5, so the 6th term must equal 3 + 5, which is 8, and the 7th term is 5 + 8, which equals 13. Now we know the first 7 terms are 1, 1, 2, 3, 5, 8, 13. Can you continue the process to find the 10th term of the sequence?)
A: 55

Q: Which table matches the graph below?
- x | y
1 | 0
2 | 2
3 | 4
4 | 6 (Incorrect. Check that the ordered pairs in each row of the table correspond to (x, y) points on the graph. Remember that the horizontal axis is represented by the x value.)
- x | y
1 | 2
2 | 3
3 | 4
4 | 6 (Incorrect. Check that the ordered pairs in each row of the table correspond to (x, y) points on the graph. Remember that the horizontal axis is represented by the x value.)
- x | y
0 | 1
2 | 2
4 | 3
6 | 4 (Correct. The points (0, 1), (2, 2), (4, 3), and (6, 4) are each points on the graph.)
- x | y
1 | 1
2 | 2
3 | 3
4 | 4 (Incorrect. Check that the ordered pairs in each row of the table correspond to (x, y) points on the graph. Remember that the horizontal axis is represented by the x value.)
A: x | y
0 | 1
2 | 2
4 | 3
6 | 4

Q: Do the following ordered pairs correspond to a function?

{(1, 5), (1, 6), (2, 7), (4, 8), (6, 9)}
- No. (Correct. These ordered pairs cannot correspond to a function since the x-value 1 corresponds to two different y-values (5 and 6).)
- Yes. (Incorrect. An output (y-value) can occur more than once, but an input (x-value) cannot correspond to more than one output. In this case, the input 1 corresponds to two different outputs (5 and 6), so the ordered pairs do not represent a function.)
A: No.

Q: Does the graph below represent a function?
- No. (Correct. The graph contains a vertical line segment. When x=-1.5, there are many corresponding y-values.)
- Yes. (Incorrect. Since the graph contains a vertical line segment when x=-1.5, there are many corresponding y-values. What does that mean?)
A: No.

Q: The graph of a function is shown below. What is its range?
- {0, 2, 5} (Incorrect. 0 and 5 are not independent values shown on the graph.)
- {1, 2, 4} (Correct. The dependent values shown in the graph are 1, 2, and 4.)
- {1, 2, 3, 5} (Incorrect. This is the domain of the function defined by the graph. 5 and 3 are not dependent values shown in the graph and 4 is a dependent value, but is not included in {1, 2, 3, 5}.)
- {1, 2, 3, 4} (Incorrect. 3 is not an independent value on this graph.)
A: {1, 2, 4}

Q: The graph below represents y as a function of x. Which statement is true about the value 4?
- 4 is in both the domain and the range of the function. (Correct. The value 4 is included at (4, -2) and (0, 4), so it is included in both the domain and the range of this function.)
- 4 is in the domain of the function, but is not in the range of the function. (Incorrect. Are there any ordered pairs with 4 as an x-value? What about with 4 as a y-value?)
- 4 is in the range of the function, but is not in the domain of the function. (Incorrect. Are there any ordered pairs with 4 as an x-value? What about with 4 as a y-value?)
- 4 is not in the domain and is not in the range of the function. (Incorrect. Are there any ordered pairs with 4 as an x-value? What about with 4 as a y-value?)
A: 4 is in both the domain and the range of the function.

Q: The table below represents y as a function of x. Which of the following statements is true about the value 4?

x | y
-1 | 0
0 | 1
2 | 3
4 | 4
5 | 7
- 4 is in the domain of the function, but is not in the range of the function. (Incorrect. Does 4 appear in the column of x-values? What about the column of y-values?)
- 4 is in both the domain and the range of the function. (Correct. 4 appears as both an x-value and a y-value in the table, so 4 is in both the domain and the range of the function.)
- 4 is in neither the domain nor the range of the function. (Incorrect. Does 4 appear in the column of x-values? What about the column of y-values?)
- 4 is in the range of the function, but is not in the domain of the function. (Incorrect. Does 4 appear in the column of x-values? What about the column of y-values?)
A: 4 is in both the domain and the range of the function.

Q: The function is a proportional function.   True or False?
- False. (Incorrect. Is there one value that you can multiply every x term by to get a corresponding y term? What does that tell you?)
- True. (Correct. In the function , every value is equal to 7 times the corresponding value, so the function is proportional.)
A: True.

Q: Sophie's favorite homemade cookie recipe requires 1 cup of chocolate chips for 6 servings. If the number of cups required for multiple batches is proportional to the number of servings being made, how many cups of chocolate chips will she need to make enough cookies for 24 servings?
- 2 (Incorrect. The relationship is proportional, and she is multiplying the number of servings by 4. She also has to multiply the amount of chocolate chips by 4.)
- 3 (Incorrect. The relationship is proportional, and she is multiplying the number of servings by 4. She also has to multiply the amount of chocolate chips by 4.)
- 4 (Correct. The relationship is proportional; if she multiplies the number of servings by 4, then she also has to multiply the amount of chocolate chips by 4.)
- 1 (Incorrect. The relationship is proportional, and she is multiplying the number of servings by 4. She also has to multiply the amount of chocolate chips by 4.)
A: 4

Q: If a function is proportional, what must the corresponding y-value equal when ?
- -1 (Incorrect. Think about some proportional functions, like and . When , what is the value of y?)
- 0 (Correct. If the function is proportional, then its equation must be of the form for some constant k. Therefore, when x=0, y equals k times 0, which equals 0.)
- 1 (Incorrect. Think about some proportional functions, like and . When , what is the value of y?)
- There is not enough information to answer this problem. (Incorrect. Think about some proportional functions, like and . When , what is the value of y?)
A: 0

Q: A 30-gallon bathtub is filled with water. A drain is opened, and the water leaves the bath in 20 minutes. If the relationship between the amount of water in the tub and the time is linear, at what rate is the tub draining?
- 1 gallon per minute (Incorrect. Since the function is linear, the number of gallons drained per minute (the function's rate of change) is constant. If 30 gallons are drained in 20 minutes, then you can represent this as the fraction . What fraction does this simplify to?)
- 1.5 gallons per minute (Correct. Since the function is linear, the number of gallons drained per minute (the function's rate of change) is constant. If 30 gallons are drained in 20 minutes, then 3 gallons will drain in 2 minutes; therefore 1.5 gallons will drain in one minute. The rate of draining is 1.5 gallons per minute.)
- gallons per minute (Incorrect. Since the function is linear, the number of gallons drained per minute (the function's rate of change) is constant. If 30 gallons are drained in 20 minutes, then you can represent this as the fraction . What fraction does this simplify to?)
- None of the above. (Incorrect. Since the function is linear, the number of gallons drained per minute (the function's rate of change) is constant. If 30 gallons are drained in 20 minutes, then you can represent this as the fraction . What fraction does this simplify to?)
A: 1.5 gallons per minute

Q: Which statement best describes the function represented by the table below?

x | y
1 | -2
2 | 1
3 | 4
4 | 7
- It is a linear function. (Correct. Since the difference between consecutive outputs is constant (each y-term is 3 greater than the previous y-term), this is a linear function.)
- It is a nonlinear function. (Incorrect. Is the difference between consecutive outputs constant or not constant?)
A: It is a linear function.

Q: An inverse variation function has the property that when . What is the equation of the function?
A: (a choice with no text)
If wrong: Incorrect. It is true that when in this function, but this is a quadratic function, not an inverse variation function.
If wrong: Incorrect. It is true that when in this function, but this is a linear function, not an inverse variation function.
If right: Correct. This function has the correct form of an inverse variation function, , and when , .
If wrong: Incorrect. It is true that when in this function, but this is a linear function, not an inverse variation function.

Q: The following partial table represents an exponential function. Which number belongs in the missing entry?

x | y
0 | 2
1 | 10
2 | 50
3 |
4 | 1250
- 72 (Incorrect. Look at the pattern in the column of y-values: 10 is 5 times larger than 2, and 50 is 5 times larger than 10. Does this help you make a prediction?)
- 60 (Incorrect. Look at the pattern in the column of y-values: 10 is 5 times larger than 2, and 50 is 5 times larger than 10. Does this help you make a prediction?)
- 650 (Incorrect. Look at the pattern in the column of y-values: 10 is 5 times larger than 2, and 50 is 5 times larger than 10. Does this help you make a prediction?)
- 250 (Correct. Since the function is exponential, the quotient of consecutive outputs is constant. Here the quotient is 5, since , and . Multiplying .)
A: 250

Q: Does this graph represent a linear, quadratic, inverse variation, or exponential function?
- This is a linear function. (Incorrect. Check the curve of the graph. What does the shape tell you about the type of function that it is?)
- This is an exponential function. (Incorrect. Check the curve of the graph. What does the shape tell you about the type of function that it is?)
- This is an inverse variation function. (Incorrect. Check the curve of the graph. What does the shape tell you about the type of function that it is?)
- This is a quadratic function. (Correct. The graph is a parabola, so the function is quadratic.)
A: This is a quadratic function.

Q: What is the best definition of "algebra"?
- The branch of mathematics that deals with operations on sets of numbers and relationships between them. (Correct. The study of algebra is concerned with operations on sets of numbers and relationships between them. Working algebraically often includes both variables and numbers.)
- The branch of mathematics that seeks to define the value of x. (Incorrect. While you may often be asked to solve for x in an algebra problem, the scope of algebra is much broader than this.)
- The branch of mathematics that uses only numbers, not variables, to represent values. (Incorrect. The branch of mathematics that uses only numbers is called arithmetic, not algebra!)
- The branch of mathematics that uses only variables, not numbers, to represent values. (Incorrect. Algebra problems do not have to include numbers, but they often do.)
A: The branch of mathematics that deals with operations on sets of numbers and relationships between them.

Q: Which of the following inequalities is not a true statement?
A: (a choice with no text)
If right: Correct. is not true; 8 is not less than or equal to 3!
If wrong: Incorrect. Recall that an inequality like can be read "a is less than b", and can be read "a is less than or equal to b." Knowing this, which of the inequalities is false?
If wrong: Incorrect. Recall that an inequality like can be read "a is less than b", and can be read "a is less than or equal to b." Knowing this, which of the inequalities is false?
If wrong: Incorrect. Recall that an inequality like can be read "a is less than b", and can be read "a is less than or equal to b." Knowing this, which of the inequalities is false?

Q: Use the order of operations to evaluate this expression:
- 18 (Incorrect. To evaluate this expression, first perform all calculations within parentheses. This means that the expression can be rewritten . Now can you multiply, then add and subtract from left to right?)
- 3 (Correct. You evaluated this expression correctly by applying the order of operations.)
- 9 (Incorrect. To evaluate this expression, first perform all calculations within parentheses. This means that the expression can be rewritten . Now can you multiply, then add and subtract from left to right?)
- 7 (Incorrect. To evaluate this expression, first perform all calculations within parentheses. This means that the expression can be rewritten . Now can you multiply, then add and subtract from left to right?)
A: 3

Q: Which of the following equations does not apply the Order of Operations correctly?
A: (a choice with no text)
If wrong: Incorrect. Remember the order of operations: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. In one of these equations, subtraction occurs before multiplication. Which one is it?
If wrong: Incorrect. Remember the order of operations: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. In one of these equations, subtraction occurs before multiplication. Which one is it?
If wrong: Incorrect. Remember the order of operations: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. In one of these equations, subtraction occurs before multiplication. Which one is it?
If right: Correct. The one that is not solved correctly is . Since multiplication should be evaluated before subtraction, , which equals 6.

Q: Which of the following equations does not apply the Order of Operations correctly?
A: (a choice with no text)
If wrong: Incorrect. Remember the order of operations: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. In one of these equations, subtraction occurs before multiplication. Which one is it?
If right: Correct. The one that is not solved correctly is . Since multiplication should be evaluated before subtraction, , which equals 0.
If wrong: Incorrect. Remember the order of operations: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. In one of these equations, subtraction occurs before multiplication. Which one is it?
If wrong: Incorrect. Remember the order of operations: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. In one of these equations, subtraction occurs before multiplication. Which one is it?

Q: The three figures below define a function with the figure number as the independent value and the number of toothpicks required to construct each figure as the dependent value.

If the domain is {1, 2, 3, 4, ...}, which of the following represents the range of the function?
- {6, 11, 16, 21, ...} (Correct. {6, 11, 16, 21, ...} is the correct answer since these values represent all of the outputs of the function.)
- {6, 11, 16, 21} (Incorrect. The range will have more than 4 numbers in it.)
- {6, 11, 16, 22, ...} (Incorrect. Figure 4 requires 21 toothpicks, so the fourth element listed in the range should be 21.)
- {6, 11, 16, 22} (Incorrect. The range will have more than 4 numbers in it. Also, figure 4 requires 21 toothpicks, so the second element listed in the range should be 21.)
A: {6, 11, 16, 21, ...}