Introductory Algebra for Middle and High School Students Practice Quiz

Answers

1. c. Each term has a base of q and an exponent of 6, so the base and exponent of your answer is q⁶.

   Add the coefficients: 12 + 4 = 16, 12q⁶ + 4q⁶ = 16q⁶.

2. a. Divide the coefficients: –45 ÷ 9 = –5. Carry the bases (a, b, and c) from the dividend into the answer. There are no bases in the divisor that are not in the dividend.

   Subtract the exponent of a in the divisor from the exponent of a in the dividend: 4 – 1 = 3. The exponent of a in the answer is 3.

   Subtract the exponent of b in the divisor from the exponent of b in the dividend: 9 – 3 = 6. The exponent of b in the answer is 6.

   Subtract the exponent of c in the divisor from the exponent of c in the dividend: 5 – 3 = 2. The exponent of c in the answer is 2.

   (–45a⁴b⁹c⁵) ÷ (9a²b³c³) = –5a³b⁶c²

3. b. Replace j with –3:

   10 + 6(–3)

   Multiply before adding:

   6(–3) = –18

   The expression becomes 10 + –18.

   Add: 10 + –18 = –8.

4. d. This expression has two g⁶ terms and two h terms.

   Combine the g⁶ terms: –7g⁶ – 8g⁶ = –15g⁶.

   Combine the h terms: 9h + 2h = 11h.

   –7g⁶ + 9h + 2h – 8g⁶ simplifies to –15g⁶ + 11h.

5. c. The keyword phrase less than signals subtraction and the keyword times signals multiplication. Three times a number is 3x. Nine less than that quantity is 3x – 9.

6. b. These terms are unlike, so they cannot be combined, but they can be factored.

   The factors of 12x⁷ are 1, 2, 3, 4, 6, 12, and x⁷, and the factors of –24x³ are 1, 2, 3, 4, 6, 8, 12, 24, and x³ (and their negatives).

   Both terms have 12 and the variable x as common factors. The smaller exponent of x between the two terms is 3, so we can factor 12x³ out of both terms.

   Divide both terms by 12x³: 12x⁷ ÷ 12x³ = x⁴, and –24x³ ÷ 12x³ = –2.

   12x⁷ – 24x³ factors into 12x³(x⁴ – 2).

7. d. 4w⁹ is raised to the third power.

   Raise 4 to the third power and raise w⁹ to the third power. 4³ = 64.

   To find (w⁹)³, multiply the exponents: (9)(3) = 27, (w⁹)³ = w²⁷, and (4w⁹)³ = 64w²⁷.

8. a. In the equation z – 7 = –9, 7 is subtracted from z. Use the opposite operation, addition, to solve the equation. Add 7 to both sides of the equation:

   z – 7 + 7 = –9 + 7

   z = –2

9. c. To find n in terms of m , we must get n alone on one side of the equation, with m on the other side of the equals sign. In the equation 4n = –28m , n is multiplied by 4. Use the opposite operation, division, to get n alone on one side of the equation. Divide both sides of the equation by 4:

   

   n = –7m

   The value of n , in terms of m , is –7m .

10. a. The equation –7k – 11 = 10 shows multiplication and subtraction. We will need to use their opposites, division and addition, to find the value of k. Add first:

    –7k – 11 + 11 = 10 + 11

    –7k = 21

    Because k is multiplied by –7, divide both sides of the equation by –7:

    

    k = –3

11. d. The equation shows division, addition, and subtraction. The variable p appears on both sides of the equation, and a constant is on both sides of the equation. Start by subtracting from both sides. This will leave us with just one p term:

    

    

    Because 2 is subtracted from , add 2 to both sides of the equation:

    

    

    Finally, because p is multiplied by , multiply both sides of the equation by :

    

    18 = p

12. b. The third root of o²⁷ is the quantity that, when multiplied three times, is equal to o²⁷. To find the third root of a base, divide the exponent of the base by 3: 27 ÷ 3 = 9. (o⁹)(o⁹)(o⁹) = o²⁷, which is why = o⁹.

13. b. The equation 8y = 16x – 4 is not in slope-intercept form, because y is not alone on one side of the equation. Since y is multiplied by 8, we must divide both sides of the equation by 8:

    

    

    The equation is now in slope-intercept form. The slope is the coefficient of x, so the slope of this line is 2.

14. a. First, find the slope. Take the difference between the first two y values in the table and divide it by the difference between the first two x values in the table: The slope of the line is 3. Next, find the y-intercept using the equation y = mx + b. Substitute 3, the slope of the line, for m. Substitute the values of x and y from the first row of the table, and solve for b, the y-intercept:

    7 = 3(1) + b

    7 = 3 + b

    4 = b

    The slope of the line is 3 and the y-intercept is 4. The equation of this line is y = 3x + 4.

15. c. The slope of the graph y = –4x – 3 is –4, since the equation is in slope intercept form and the coefficient of x is –4. The y-intercept of the equation is –3. The graphs in choices a and c have y-intercepts of –3, but only the graph in choice c has a slope of –4.

16. d. To find the equation of the line, begin by finding the slope using any two points on the line. When x = 1, y = 6, and when x = 2, y = 5. The slope is equal to: The y-intercept can be found right on the graph. The line crosses the y-axis where y = 7, which means that 7 is the y-intercept. This is the graph of the equation y = –x + 7.

17. c. Use the distance formula to find the distance between (–3,–3) and (9,13). Because (–3,–3) is the first point, x₁ will be –3 and y₁ will be –3. (9,13) is the second point, so x₂ will be 9 and y₂ will be 13:

    

18. b. An equation is a function if every x value has no more than one y value. In the equation x = y², every positive x value will have two y values, since the square of a positive number and the square of its negative are the same. The equation x = y² is not a function.

19. a. This system could be solved using either substitution or elimination. Because the first equation can easily be solved for x, use substitution to solve.

    Write x in terms of y by subtracting 4y from both sides of the first equation:

    4y + x = 7

    x = 7 – 4y

    Replace x in the second equation with the expression that is equal to x, 7 – 4y:

    3y – 2(7 – 4y) = 30

    Solve for the value of y:

    3y – 2(7 – 4y) = 30

    3y – 14 + 8y = 30

    11y – 14 = 30

    11y = 44

    y = 4

    Replace y with its value in either equation and solve for the value of x:

    4(4) + x = 7

    16 + x = 7

    x = –9

20. d. The graph shows the number –4 circled, and the circle is solid. This means that –4 is part of the solution set. All of the values that are greater than –4 are highlighted, which means that the solution set is all values that are greater than or equal to –4: x ≥ –4.

21. a. The solution set of two inequalities is the area of a graph where the solution set of one inequality overlaps the solution set of the other inequality.

    The line y = x is dashed, which means that the points on the line are not part of the solution set.

    The line y = –4x is solid, which means that the points on the line are part of the solution set.

    The point (–4, 4) is in the solution set, so it can be used to test each inequality.

    (–4) = –1, which is less than 4. Therefore, y > x.

    –4(–4) = 16, which is greater than 4. Therefore, y ≤ –4x.

    This graph shows the solution set of y > x or y ≤ –4x.

22. b. To find the product of two binomials, use FOIL and combine like terms:

    First: (7x)(x) = 7x²

    Outside: (7x)(2) = 14x

    Inside: (–1)(x) = –x

    Last: (–1)(2) = –2

    7x² + 14x – x – 2 = 7x² + 13x – 2

23. d. To factor x² + 10x + 24, begin by listing the positive and negative factors of the first and last terms:

    Factors of x²: 1, –x, x, x²

    Factors of 24: –24, –12, –8, –6, –4, –3, –2, –1, 1, 2, 3, 4, 6, 8, 12, 24

    x² is the square of either x or –x. Begin by trying x as the first term in each binomial:

    (x + __)(x + __)

    The coefficients of each x term are 1. The last terms of each binomial must multiply to 24 and add to 10, since 10x is the sum of the Outside and Inside products. 10x is positive, so we are looking for two positive numbers that multiply to 24 and add to 10.

    (2)(12) = 24, but 2 + 12 = 14

    (3)(8) = 24, but 3 + 8 = 11

    (4)(6) = 24, and 4 + 6 = 10

    The constant of one binomial is 4 and the constant of the other binomial is 6:

    (x + 4)(x + 6)

    Check the answer using FOIL:

    First: (x)(x) = x²

    Outside: (x)(6) = 6x

    Inside: (4)(x) = 4x

    Last: (4)(6) = 24

    x² + 6x + 4x + 24 = x² + 10x + 24

24. c. Before we can find the roots of x² + 6x = 7, it must be in the form ax² + bx + c = 0. Subtract 7 from both sides of the equation. The equation is now x² + 6x – 7 = 0.

    Factor x² + 6x – 7 and set each factor equal to 0.

    Factors of x²: 1, –x, x, x²

    Factors of –7: –7, –1, 1, 7

    Because the last term is –7, we are looking for a negative number and a positive number that multiply to –7:

    (x – 7)(x + 1) = x² – 6x – 7. The middle term is too small.

    (x + 7)(x – 1) = x² + 6x – 7. These are the correct factors./ul>

    Set x + 7 and x – 1 equal to 0 and solve for x:

    x + 7 = 0

    x – 1 = 0

    x = –7

    x = 1

    The roots of x² + 6x – 7 = 0 are –7 and 1.

25. d. We are looking for Jamie's score, so we can use x to represent that number. Lindsay's score is seven less than three times that number, which means that her score is 3x – 7. We know that her score is 134, so we can set 3x – 7 equal to 134 and solve for x:

    3x – 7 = 134

    3x = 141

    x = 47

    Jamie's score is 47.

26. c. Write the ratio of yellow beads to green beads as a fraction. 3:10 = The number of green beads is unknown, so represent that number with x. The ratio of actual yellow beads to actual green beads is 60:x, or Set these ratios equal to each other, cross multiply, and solve for x:

    

    3x = 600

    x = 200

    There are 200 green beads on the necklace.

27. b. The mean of a set is equal to the sum of the values divided by the number of values. Let x represent the seventh value in the set:

    9 + 19 + 17 + 16 + 11 + 20 + x = 92 + x

    

    92 + x = 105

    x = 13

    The seventh value of the set is 13.

28. d. The original value is 48 and the new value is x. Percent increase is equal to the new value minus the original value divided by the original value.

    Subtract 48 from x and divide by 48. Set that fraction equal to 55%, which is 0.55:

    

    Multiply both sides of the equation by 48, and then add 48 to both sides:

    x – 48 = 26.4

    x = 74.4

    48 after a 55% increase is 74.4.

29. a. The formula t_n = t₁ + (n – 1)d gives us the value of any term in an arithmetic sequence. The first term, t₁, is –3x + 1.

    The difference, d, can be found by subtracting any pair of consecutive terms: x – 7 – (–3x + 1) = 4x – 8. d = 4x – 8. Because we're looking for the ninth term, n = 9:

    t₉ = –3x + 1 + (9 – 1)(4x – 8)

    t₉ = –3x + 1 + (8)(4x – 8)

    t₉ = –3x + 1 + 32x – 64

    t₉ = 29x – 63

    The ninth term in the sequence is 29x – 63.

    The difference between any two consecutive terms in an arithmetic sequence is always the same. The difference between the first term and the second term is 4x – 8. The difference between the second term and the third term is 2x – 3 – (x – 7) = x + 4.

    Set these two differences equal to each other and solve for x:

    4x – 8 = x + 4

    3x – 8 = 4

    3x = 12

    x = 4

    The value of x in this sequence is 4, which means that the ninth term, 29x – 63, is equal to 29(4) – 63 = 116 – 63 = 53.

30. c. The formula for perimeter of a rectangle is P = 2l + 2w. Let x represent the length of the rectangle. The width is 4 times the length, which means that it is 4x. Substitute 560 for P, x for l, and 4x for w:

    560 = 2(x) + 2(4x)

    560 = 2x + 8x

    560 = 10x

    56 = x

    Because x = 56, the length of the rectangle is 56 m. The width is four times the length, which means that the width is 4(56) = 224 m.