Algebra practice questions

Algebra questions

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1. 1Linear equationsFoundation

   If 3x + 5 = 23, what is the value of x?If 3x + 5 = 23, what is the value of x?

   • A66
   • B44
   • C2828
   • D88
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   Answer: 66

   Subtract 5 from both sides to get 3x = 18. Divide by 3, so x = 6.Subtract 5 from both sides to get 3x = 18. Divide by 3, so x = 6.

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2. 2Linear functionsFoundation

   For the line y = 2x - 3, what is the value of y when x = 4?For the line y = 2x - 3, what is the value of y when x = 4?

   • A88
   • B22
   • C1111
   • D55
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   Answer: 55

   Substitute 4 for x: y = 2(4) - 3 = 8 - 3 = 5. Evaluating a line at a point is only substitution — there is nothing to solve.Substitute 4 for x: y = 2(4) - 3 = 8 - 3 = 5. Evaluating a line at a point is only substitution — there is nothing to solve.

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3. 3Systems of equationsStandard

   If x + y = 10 and x - y = 4, what is the value of x?If x + y = 10 and x - y = 4, what is the value of x?

   • A55
   • B33
   • C1414
   • D77
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   Answer: 77

   The y terms are already opposites, so adding the equations eliminates y: 2x = 14, and x = 7. Check for that before substituting; it is faster whenever a variable appears with matching coefficients.The y terms are already opposites, so adding the equations eliminates y: 2x = 14, and x = 7. Check for that before substituting; it is faster whenever a variable appears with matching coefficients.

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4. 4Linear equationsStandard

   What is the solution to 5(x - 2) = 3x + 10?What is the solution to 5(x - 2) = 3x + 10?

   • A55
   • B1515
   • C2020
   • D1010
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   Answer: 1010

   Expand first: 5x - 10 = 3x + 10. Subtract 3x and add 10: 2x = 20, so x = 10.Expand first: 5x - 10 = 3x + 10. Subtract 3x and add 10: 2x = 20, so x = 10.

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5. 5Function notationFoundation

   If f(x) = 4x + 1 and f(t) = 21, what is the value of t?If f(x) = 4x + 1 and f(t) = 21, what is the value of t?

   • A2020
   • B55
   • C44
   • D66
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   Answer: 55

   f(t) = 21 means 4t + 1 = 21. Then 4t = 20, so t = 5. Function notation is only naming: replace f(t) with the rule and solve the equation you already know how to solve.f(t) = 21 means 4t + 1 = 21. Then 4t = 20, so t = 5. Function notation is only naming: replace f(t) with the rule and solve the equation you already know how to solve.

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6. 6Linear equationsFoundation

   What is the value of x in x + 34 = 5?What is the value of x in (x + 3)/4 = 5?

   • A77
   • B1717
   • C2020
   • D2323
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   Answer: 1717

   Multiply both sides by 4: x + 3 = 20. Subtract 3 to get x = 17.Multiply both sides by 4: x + 3 = 20. Subtract 3 to get x = 17.

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7. 7Proportional relationshipsFoundation

   If 0.4x = 18, what is x?If 0.4x = 18, what is x?

   • A7.27.2
   • B22.522.5
   • C4545
   • D7272
   Show the answer and explanationHide the explanation

   Answer: 4545

   Divide both sides by the coefficient: x = 180.4 = 45. Dividing by a decimal below 1 makes the result larger, which is the check that rules out 7.2.Divide both sides by the coefficient: x = 18/0.4 = 45. Dividing by a decimal below 1 makes the result larger, which is the check that rules out 7.2.

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8. 8Linear inequalitiesFoundation

   Which inequality is equivalent to 2x - 7 > 5?Which inequality is equivalent to 2x - 7 > 5?

   • Ax < 1x < 1
   • Bx < 6x < 6
   • Cx > 1x > 1
   • Dx > 6x > 6
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   Answer: x > 6x > 6

   Add 7 to both sides to get 2x > 12. Divide by positive 2, so x > 6.Add 7 to both sides to get 2x > 12. Divide by positive 2, so x > 6.

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9. 9Direct variationStandard

   In a direct variation, y = kx. If y = 12 when x = 3, what is y when x = 7?In a direct variation, y = kx. If y = 12 when x = 3, what is y when x = 7?

   • A2121
   • B1616
   • C8484
   • D2828
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   Answer: 2828

   Find k from the pair you are given, then use it: 12 = 3k gives k = 4, and y = 4(7) = 28. Every direct variation question is these two steps in this order.Find k from the pair you are given, then use it: 12 = 3k gives k = 4, and y = 4(7) = 28. Every direct variation question is these two steps in this order.

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10. 10Linear equationsFoundation

    What is the solution to 4 - 2x = 10?What is the solution to 4 - 2x = 10?

    • A33
    • B-3-3
    • C22
    • D-2-2
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    Answer: -3-3

    Subtract 4 from both sides: -2x = 6. Divide by -2, so x = -3.Subtract 4 from both sides: -2x = 6. Divide by -2, so x = -3.

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11. 11Equivalent expressionsFoundation

    Which expression is equivalent to 3(x + 2) - 2?Which expression is equivalent to 3(x + 2) - 2?

    • A3x + 43x + 4
    • B3x - 43x - 4
    • Cx + 4x + 4
    • D3x + 63x + 6
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    Answer: 3x + 43x + 4

    Distribute first, then collect: 3(x + 2) = 3x + 6, and 3x + 6 - 2 = 3x + 4. The usual slip is distributing to the first term only and leaving 3x + 2 - 2.Distribute first, then collect: 3(x + 2) = 3x + 6, and 3x + 6 - 2 = 3x + 4. The usual slip is distributing to the first term only and leaving 3x + 2 - 2.

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12. 12SlopeFoundation

    What is the slope of the line passing through (1, 4) and (3, 10)?What is the slope of the line passing through (1, 4) and (3, 10)?

    • A22
    • B66
    • C44
    • D33
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    Answer: 33

    Slope is the change in y over the change in x: 10 - 43 - 1 = 62 = 3.Slope is the change in y over the change in x: (10 - 4)/(3 - 1) = 6/2 = 3.

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13. 13Linear equationsStandard

    What is the solution to 2x3 - 1 = 5?What is the solution to (2x)/3 - 1 = 5?

    • A99
    • B1212
    • C88
    • D66
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    Answer: 99

    Add 1 to both sides to get 2x / 3 = 6. Multiply by 3 to get 2x = 18, so x = 9.Add 1 to both sides to get 2x / 3 = 6. Multiply by 3 to get 2x = 18, so x = 9.

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14. 14Systems of equationsStandard

    If y = 3x - 2 and 2x + y = 13, what is the value of x?If y = 3x - 2 and 2x + y = 13, what is the value of x?

    • A55
    • B44
    • C33
    • D22
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    Answer: 33

    Substitute 3x - 2 for y: 2x + 3x - 2 = 13, so 5x = 15 and x = 3.Substitute 3x - 2 for y: 2x + 3x - 2 = 13, so 5x = 15 and x = 3.

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15. 15Linear inequalitiesChallenge

    Which inequality is equivalent to -3x + 4 ≤ 19?Which inequality is equivalent to -3x + 4 ≤ 19?

    • Ax ≥ -5x ≥ -5
    • Bx ≤ 5x ≤ 5
    • Cx ≤ -5x ≤ -5
    • Dx ≥ 5x ≥ 5
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    Answer: x ≥ -5x ≥ -5

    Subtract 4 to get -3x ≤ 15. Dividing by a negative number reverses the inequality, so x ≥ -5.Subtract 4 to get -3x ≤ 15. Dividing by a negative number reverses the inequality, so x ≥ -5.

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16. 16Linear functionsFoundation

    A line passes through (0, 7) and has slope -2. What is the value of y when x = 4?A line passes through (0, 7) and has slope -2. What is the value of y when x = 4?

    • A11
    • B1515
    • C-1-1
    • D-15-15
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    Answer: -1-1

    The line is y = -2x + 7. Substituting x = 4 gives y = -8 + 7 = -1.The line is y = -2x + 7. Substituting x = 4 gives y = -8 + 7 = -1.

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17. 17Function notationChallenge

    If f(x) = 2x - 5, what is the value of f(f(3))?If f(x) = 2x - 5, what is the value of f(f(3))?

    • A-1-1
    • B33
    • C-3-3
    • D11
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    Answer: -3-3

    Work from the inside out: f(3) = 6 - 5 = 1, then f(1) = 2 - 5 = -3. The output of the first becomes the input of the second; f(f(3)) is not f(3) doubled.Work from the inside out: f(3) = 6 - 5 = 1, then f(1) = 2 - 5 = -3. The output of the first becomes the input of the second; f(f(3)) is not f(3) doubled.

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18. 18Proportional relationshipsStandard

    A recipe uses 5 cups of flour to make 2 loaves of bread. At the same rate, enter the number of cups needed for 7 loaves.A recipe uses 5 cups of flour to make 2 loaves of bread. At the same rate, enter the number of cups needed for 7 loaves.

    Student-produced response: write your own answer rather than choosing one.

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    Answer: 17.517.5

    The rate is 52 = 2.5 cups per loaf. For 7 loaves, 2.5 × 7 = 17.5 cups.The rate is 5/2 = 2.5 cups per loaf. For 7 loaves, 2.5 × 7 = 17.5 cups.

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19. 19Equivalent expressionsStandard

    If 3a + 2b = 12, which expression gives a in terms of b?If 3a + 2b = 12, which expression gives a in terms of b?

    • A12 + 2b3(12 + 2b)/3
    • B12 - 2b12 - 2b
    • C12 - 2b3(12 - 2b)/3
    • D2b - 123(2b - 12)/3
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    Answer: 12 - 2b3(12 - 2b)/3

    Subtract 2b from both sides to get 3a = 12 - 2b, then divide both sides by 3.Subtract 2b from both sides to get 3a = 12 - 2b, then divide both sides by 3.

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20. 20Systems of equationsChallenge

    For which value of k does the system 2x + ky = 8 and 4x + 6y = 5 have no solution?For which value of k does the system 2x + ky = 8 and 4x + 6y = 5 have no solution?

    • A22
    • B1212
    • C33
    • D66
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    Answer: 33

    A system has no solution when the two lines are parallel but not identical. Multiplying the first equation by 2 gives 4x + 2ky = 16, which matches 4x + 6y in the x and y terms when 2k = 6, so k = 3. The constants 16 and 5 differ, so the lines never meet.A system has no solution when the two lines are parallel but not identical. Multiplying the first equation by 2 gives 4x + 2ky = 16, which matches 4x + 6y in the x and y terms when 2k = 6, so k = 3. The constants 16 and 5 differ, so the lines never meet.

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21. 21Linear equationsFoundation

    What is the solution to 2(3x - 1) = 4x + 8?What is the solution to 2(3x - 1) = 4x + 8?

    • A55
    • B33
    • C44
    • D66
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    Answer: 55

    Distribute to get 6x - 2 = 4x + 8. Subtract 4x and add 2 to get 2x = 10, so x = 5.Distribute to get 6x - 2 = 4x + 8. Subtract 4x and add 2 to get 2x = 10, so x = 5.

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22. 22Linear functionsFoundation

    The cost C, in dollars, of renting a bicycle for h hours is C = 4 + 3h. Enter the number of hours that gives a cost of 25 dollars.The cost C, in dollars, of renting a bicycle for h hours is C = 4 + 3h. Enter the number of hours that gives a cost of 25 dollars.

    Student-produced response: write your own answer rather than choosing one.

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    Answer: 77

    Set the rule equal to the total and solve: 4 + 3h = 25, so 3h = 21 and h = 7 hours. The 4 is a fixed charge, not part of the hourly rate, so it comes off before dividing.Set the rule equal to the total and solve: 4 + 3h = 25, so 3h = 21 and h = 7 hours. The 4 is a fixed charge, not part of the hourly rate, so it comes off before dividing.

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23. 23Linear functionsStandard

    A line has slope 4 and passes through (2, 3). Which equation describes the line?A line has slope 4 and passes through (2, 3). Which equation describes the line?

    • Ay = 4x + 3y = 4x + 3
    • By - 3 = 4(x - 2)y - 3 = 4(x - 2)
    • Cy + 3 = 4(x + 2)y + 3 = 4(x + 2)
    • Dy - 2 = 4(x - 3)y - 2 = 4(x - 3)
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    Answer: y - 3 = 4(x - 2)y - 3 = 4(x - 2)

    Point-slope form is y - y₁ = m(x - x₁). Substituting m = 4 and the point (2, 3) gives y - 3 = 4(x - 2). Note that the point's coordinates are subtracted, so both signs are negative here.Point-slope form is y - y1 = m(x - x1). Substituting m = 4 and the point (2, 3) gives y - 3 = 4(x - 2). Note that the point's coordinates are subtracted, so both signs are negative here.

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24. 24Linear functionsChallenge

    A line is perpendicular to y = -2x + 5 and passes through the origin. What is its equation?A line is perpendicular to y = -2x + 5 and passes through the origin. What is its equation?

    • Ay = (12)xy = (1/2)x
    • By = 2xy = 2x
    • Cy = -2xy = -2x
    • Dy = -(12)xy = -(1/2)x
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    Answer: y = (12)xy = (1/2)x

    Perpendicular slopes are negative reciprocals: the negative reciprocal of -2 is 12. A line through the origin has no constant term, so y = (12)x.Perpendicular slopes are negative reciprocals: the negative reciprocal of -2 is 1/2. A line through the origin has no constant term, so y = (1/2)x.

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25. 25Absolute valueStandard

    How many values of x satisfy the equation |x - 4| = 7?How many values of x satisfy the equation |x - 4| = 7?

    • A22
    • BInfinitely manyInfinitely many
    • C00
    • D11
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    Answer: 22

    An absolute value equals 7 when the expression inside is 7 or -7. Solving x - 4 = 7 gives x = 11 and x - 4 = -7 gives x = -3, so there are two solutions.An absolute value equals 7 when the expression inside is 7 or -7. Solving x - 4 = 7 gives x = 11 and x - 4 = -7 gives x = -3, so there are two solutions.

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26. 26Interpreting linear modelsFoundation

    A phone plan costs C = 15 + 0.05m dollars, where m is the number of minutes used. What does the 0.05 represent?A phone plan costs C = 15 + 0.05m dollars, where m is the number of minutes used. What does the 0.05 represent?

    • AThe total cost of the planThe total cost of the plan
    • BThe number of free minutesThe number of free minutes
    • CThe fixed monthly feeThe fixed monthly fee
    • DThe cost of each additional minuteThe cost of each additional minute
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    Answer: The cost of each additional minuteThe cost of each additional minute

    In a linear model, the coefficient of the variable is the rate of change. Here each extra minute adds $0.05, while the 15 is the fixed fee that applies regardless of usage.In a linear model, the coefficient of the variable is the rate of change. Here each extra minute adds $0.05, while the 15 is the fixed fee that applies regardless of usage.

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27. 27Systems of equationsStandard

    If 3x + 2y = 16 and 3x - 2y = 4, what is the value of y?If 3x + 2y = 16 and 3x - 2y = 4, what is the value of y?

    • A66
    • B33
    • C44
    • D22
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    Answer: 33

    Subtract the second equation from the first: (3x + 2y) - (3x - 2y) = 16 - 4 gives 4y = 12, so y = 3. Elimination is cheaper here than substitution because the x terms already match.Subtract the second equation from the first: (3x + 2y) - (3x - 2y) = 16 - 4 gives 4y = 12, so y = 3. Elimination is cheaper here than substitution because the x terms already match.

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28. 28Linear inequalitiesStandard

    A delivery van can carry at most 900 kilograms. It is already loaded with 240 kilograms. If each crate weighs 30 kilograms, what is the greatest number of crates it can still take?A delivery van can carry at most 900 kilograms. It is already loaded with 240 kilograms. If each crate weighs 30 kilograms, what is the greatest number of crates it can still take?

    • A3030
    • B2323
    • C2020
    • D2222
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    Answer: 2222

    The remaining capacity is 900 - 240 = 660 kilograms. Dividing by 30 gives 22 crates exactly, and 23 crates would exceed the limit.The remaining capacity is 900 - 240 = 660 kilograms. Dividing by 30 gives 22 crates exactly, and 23 crates would exceed the limit.

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29. 29Function notationStandard

    If f(x) = 5 - 2x, for what value of x does f(x) = f(1) + 6?If f(x) = 5 - 2x, for what value of x does f(x) = f(1) + 6?

    • A22
    • B-3-3
    • C-2-2
    • D44
    Show the answer and explanationHide the explanation

    Answer: -2-2

    First f(1) = 5 - 2 = 3, so the target value is 3 + 6 = 9. Solving 5 - 2x = 9 gives -2x = 4 and x = -2.First f(1) = 5 - 2 = 3, so the target value is 3 + 6 = 9. Solving 5 - 2x = 9 gives -2x = 4 and x = -2.

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30. 30Linear equationsFoundation

    What is the solution to 7 - 3(x - 1) = 1?What is the solution to 7 - 3(x - 1) = 1?

    • A33
    • B22
    • C11
    • D-3-3
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    Answer: 33

    Distribute carefully: 7 - 3x + 3 = 1, so 10 - 3x = 1. Then -3x = -9 and x = 3. Writing 7 - 3x - 3 is the common slip.Distribute carefully: 7 - 3x + 3 = 1, so 10 - 3x = 1. Then -3x = -9 and x = 3. Writing 7 - 3x - 3 is the common slip.

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31. 31Proportional relationshipsChallenge

    If y varies directly with x and y = 18 when x = 4, enter the value of x when y = 45.If y varies directly with x and y = 18 when x = 4, enter the value of x when y = 45.

    Student-produced response: write your own answer rather than choosing one.

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    Answer: 1010

    Direct variation means y = kx with a constant k. From 18 = 4k, k = 4.5. Then 45 = 4.5x gives x = 10.Direct variation means y = kx with a constant k. From 18 = 4k, k = 4.5. Then 45 = 4.5x gives x = 10.

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32. 32Interpreting linear modelsStandard

    The number of members of a club is modelled by M = 48 + 7t, where t is years since opening. What was the membership at opening, and how fast is it growing?The number of members of a club is modelled by M = 48 + 7t, where t is years since opening. What was the membership at opening, and how fast is it growing?

    • A7 members, growing by 48 a year7 members, growing by 48 a year
    • B48 members, growing by 7 a year48 members, growing by 7 a year
    • C48 members, growing by 48 a year48 members, growing by 48 a year
    • D55 members, growing by 7 a year55 members, growing by 7 a year
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    Answer: 48 members, growing by 7 a year48 members, growing by 7 a year

    At t = 0 the model gives M = 48, which is the starting value. The coefficient of t is the annual rate of change, so membership grows by 7 each year.At t = 0 the model gives M = 48, which is the starting value. The coefficient of t is the annual rate of change, so membership grows by 7 each year.

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33. 33Linear equationsStandard

    If 3x - 12 = x + 73, what is the value of x?If (3x - 1)/2 = (x + 7)/3, what is the value of x?

    • A55
    • B17717/7
    • C171117/11
    • D33
    Show the answer and explanationHide the explanation

    Answer: 17717/7

    Cross-multiply: 3(3x - 1) = 2(x + 7), so 9x - 3 = 2x + 14. Then 7x = 17 and x = 177.Cross-multiply: 3(3x - 1) = 2(x + 7), so 9x - 3 = 2x + 14. Then 7x = 17 and x = 17/7.

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34. 34Linear inequalitiesChallenge

    Which values of x satisfy both 2x + 1 > 7 and 5 - x > 0?Which values of x satisfy both 2x + 1 > 7 and 5 - x > 0?

    • AThere are no such values.There are no such values.
    • B3 < x < 53 < x < 5
    • Cx < 5x < 5
    • Dx > 3x > 3
    Show the answer and explanationHide the explanation

    Answer: 3 < x < 53 < x < 5

    The first gives 2x > 6, so x > 3. The second gives x < 5. Both hold only between them, which is 3 < x < 5.The first gives 2x > 6, so x > 3. The second gives x < 5. Both hold only between them, which is 3 < x < 5.

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35. 35Systems of equationsChallenge

    The system 3x + ky = 9 and 6x + 10y = 18 has infinitely many solutions. What is k?The system 3x + ky = 9 and 6x + 10y = 18 has infinitely many solutions. What is k?

    • A1010
    • B55
    • C33
    • D2020
    Show the answer and explanationHide the explanation

    Answer: 55

    Infinitely many solutions means the equations describe the same line. Doubling the first gives 6x + 2ky = 18, which matches the second when 2k = 10, so k = 5. The constants already agree, which is what makes them identical rather than parallel.Infinitely many solutions means the equations describe the same line. Doubling the first gives 6x + 2ky = 18, which matches the second when 2k = 10, so k = 5. The constants already agree, which is what makes them identical rather than parallel.

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36. 36Function notationStandard

    If f(x) = 3x + 2 and g(x) = x - 4, what is f(g(5))?If f(x) = 3x + 2 and g(x) = x - 4, what is f(g(5))?

    • A55
    • B1111
    • C1313
    • D1717
    Show the answer and explanationHide the explanation

    Answer: 55

    Work from the inside out: g(5) = 1, then f(1) = 3 + 2 = 5. Applying f first would give a different and incorrect answer.Work from the inside out: g(5) = 1, then f(1) = 3 + 2 = 5. Applying f first would give a different and incorrect answer.

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37. 37Proportional relationshipsFoundation

    A machine fills 250 bottles in 20 minutes at a constant rate. Enter the number of bottles it fills in 50 minutes.A machine fills 250 bottles in 20 minutes at a constant rate. Enter the number of bottles it fills in 50 minutes.

    Student-produced response: write your own answer rather than choosing one.

    Show the answer and explanationHide the explanation

    Answer: 625625

    The rate is 25020 = 12.5 bottles per minute, and 12.5 × 50 = 625 bottles.The rate is 250/20 = 12.5 bottles per minute, and 12.5 × 50 = 625 bottles.

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38. 38Equivalent expressionsStandard

    Which expression is equivalent to 2(x - 3) - 3(2x + 1)?Which expression is equivalent to 2(x - 3) - 3(2x + 1)?

    • A-4x - 9-4x - 9
    • B-4x + 9-4x + 9
    • C8x - 98x - 9
    • D-4x - 3-4x - 3
    Show the answer and explanationHide the explanation

    Answer: -4x - 9-4x - 9

    Distribute both: 2x - 6 - 6x - 3. Note the -3 multiplies both terms in the second bracket, giving -3 rather than +3. Collecting gives -4x - 9.Distribute both: 2x - 6 - 6x - 3. Note the -3 multiplies both terms in the second bracket, giving -3 rather than +3. Collecting gives -4x - 9.

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39. 39Linear functionsChallenge

    A line passes through (-2, 7) and (4, -5). At what value of x does it cross the x-axis?A line passes through (-2, 7) and (4, -5). At what value of x does it cross the x-axis?

    • A3.03.0
    • B1.51.5
    • C-1.5-1.5
    • D2.02.0
    Show the answer and explanationHide the explanation

    Answer: 1.51.5

    The slope is -5 - 74 - (-2) = -126 = -2. Using (4, -5): -5 = -2(4) + b gives b = 3, so y = -2x + 3. Setting y = 0 gives x = 1.5.The slope is (-5 - 7)/(4 - (-2)) = -12/6 = -2. Using (4, -5): -5 = -2(4) + b gives b = 3, so y = -2x + 3. Setting y = 0 gives x = 1.5.

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40. 40Absolute valueStandard

    Which inequality describes all values within 4 units of 9 on the number line?Which inequality describes all values within 4 units of 9 on the number line?

    • A|x - 4| ≤ 9|x - 4| ≤ 9
    • B|9 - 4| ≤ x|9 - 4| ≤ x
    • C|x + 9| ≤ 4|x + 9| ≤ 4
    • D|x - 9| ≤ 4|x - 9| ≤ 4
    Show the answer and explanationHide the explanation

    Answer: |x - 9| ≤ 4|x - 9| ≤ 4

    Distance from 9 is written |x - 9|, and within 4 units means that distance is at most 4. The subtraction names the center, not the radius.Distance from 9 is written |x - 9|, and within 4 units means that distance is at most 4. The subtraction names the center, not the radius.

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41. 41Linear equationsFoundation

    If 8 - 2x = 3x - 7, what is the value of x?If 8 - 2x = 3x - 7, what is the value of x?

    • A-3-3
    • B1515
    • C33
    • D11
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    Answer: 33

    Collect the variable on the side that keeps it positive: adding 2x gives 8 = 5x - 7, then adding 7 gives 15 = 5x, so x = 3. Choosing that side first avoids a sign error later.Collect the variable on the side that keeps it positive: adding 2x gives 8 = 5x - 7, then adding 7 gives 15 = 5x, so x = 3. Choosing that side first avoids a sign error later.

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42. 42Systems of equationsStandard

    If 2x + 5y = 26 and x = y + 1, what is the value of y?If 2x + 5y = 26 and x = y + 1, what is the value of y?

    • A26726/7
    • B33
    • C24724/7
    • D44
    Show the answer and explanationHide the explanation

    Answer: 24724/7

    Substitute x = y + 1: 2(y + 1) + 5y = 26, so 7y + 2 = 26 and 7y = 24, giving y = 247.Substitute x = y + 1: 2(y + 1) + 5y = 26, so 7y + 2 = 26 and 7y = 24, giving y = 24/7.

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43. 43Linear inequalitiesStandard

    A taxi charges a $3 flag fee plus $1.80 per mile. What is the greatest whole number of miles that can be traveled for $30?A taxi charges a $3 flag fee plus $1.80 per mile. What is the greatest whole number of miles that can be traveled for $30?

    • A1515
    • B1717
    • C1414
    • D1616
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    Answer: 1515

    Solve 3 + 1.80m ≤ 30, so 1.80m ≤ 27 and m ≤ 15. Exactly 15 miles costs $30, and 16 would exceed it.Solve 3 + 1.80m ≤ 30, so 1.80m ≤ 27 and m ≤ 15. Exactly 15 miles costs $30, and 16 would exceed it.

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44. 44Function notationChallenge

    If f(x) = ax + 3 and f(2) = 11, what is f(5)?If f(x) = ax + 3 and f(2) = 11, what is f(5)?

    • A2626
    • B2020
    • C1919
    • D2323
    Show the answer and explanationHide the explanation

    Answer: 2323

    From f(2) = 11: 2a + 3 = 11, so a = 4 and f(x) = 4x + 3. Then f(5) = 20 + 3 = 23.From f(2) = 11: 2a + 3 = 11, so a = 4 and f(x) = 4x + 3. Then f(5) = 20 + 3 = 23.

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45. 45Proportional relationshipsStandard

    If 5 workers build a wall in 12 days, enter how many days 4 workers would take at the same rate.If 5 workers build a wall in 12 days, enter how many days 4 workers would take at the same rate.

    Student-produced response: write your own answer rather than choosing one.

    Show the answer and explanationHide the explanation

    Answer: 1515

    The total work is 5 × 12 = 60 worker-days. Dividing by 4 workers gives 15 days. Fewer workers means more days, so the numbers move in opposite directions.The total work is 5 × 12 = 60 worker-days. Dividing by 4 workers gives 15 days. Fewer workers means more days, so the numbers move in opposite directions.

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46. 46Equivalent expressionsStandard

    Which expression is equivalent to (6x² + 9x) / 3x, for x not equal to 0?Which expression is equivalent to (6x^2 + 9x) / 3x, for x not equal to 0?

    • A6x + 36x + 3
    • B2x² + 32x^2 + 3
    • C2x + 92x + 9
    • D2x + 32x + 3
    Show the answer and explanationHide the explanation

    Answer: 2x + 32x + 3

    Divide each term by 3x: 6x² / 3x = 2x and 9x / 3x = 3. Dividing only the first term is the common slip.Divide each term by 3x: 6x^2 / 3x = 2x and 9x / 3x = 3. Dividing only the first term is the common slip.

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47. 47Linear functionsChallenge

    Two lines are perpendicular. One has equation y = (34)x - 2. What is the slope of the other?Two lines are perpendicular. One has equation y = (3/4)x - 2. What is the slope of the other?

    • A-34-3/4
    • B-43-4/3
    • C343/4
    • D434/3
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    Answer: -43-4/3

    Perpendicular slopes are negative reciprocals: flip 34 to 43 and change the sign, giving -43.Perpendicular slopes are negative reciprocals: flip 3/4 to 4/3 and change the sign, giving -4/3.

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48. 48Absolute valueChallenge

    How many values of x satisfy |2x - 6| = -4?How many values of x satisfy |2x - 6| = -4?

    • A11
    • BInfinitely manyInfinitely many
    • C00
    • D22
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    Answer: 00

    An absolute value is never negative, so no value of x can make it equal -4. Solving the two cases anyway produces answers that fail when checked.An absolute value is never negative, so no value of x can make it equal -4. Solving the two cases anyway produces answers that fail when checked.

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49. 49Linear equationsStandard

    If 4(x + 2) - 3 = 2(x + 9), what is the value of x?If 4(x + 2) - 3 = 2(x + 9), what is the value of x?

    • A55
    • B88
    • C6.56.5
    • D3.53.5
    Show the answer and explanationHide the explanation

    Answer: 6.56.5

    Distribute both sides: 4x + 8 - 3 = 2x + 18, so 4x + 5 = 2x + 18. Then 2x = 13 and x = 6.5.Distribute both sides: 4x + 8 - 3 = 2x + 18, so 4x + 5 = 2x + 18. Then 2x = 13 and x = 6.5.

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50. 50Systems of equationsStandard

    If 4x - y = 11 and x + y = 4, what is the value of x?If 4x - y = 11 and x + y = 4, what is the value of x?

    • A55
    • B33
    • C22
    • D11
    Show the answer and explanationHide the explanation

    Answer: 33

    Adding the equations eliminates y: 5x = 15, so x = 3. Elimination is cheaper here than substitution because the y terms already cancel.Adding the equations eliminates y: 5x = 15, so x = 3. Elimination is cheaper here than substitution because the y terms already cancel.

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51. 51Linear functionsStandard

    A line has x-intercept 4 and y-intercept -6. What is its slope?A line has x-intercept 4 and y-intercept -6. What is its slope?

    • A232/3
    • B-23-2/3
    • C-1.5-1.5
    • D1.51.5
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    Answer: 1.51.5

    The line passes through (4, 0) and (0, -6). Slope is 0 - (-6)4 - 0 = 64 = 1.5.The line passes through (4, 0) and (0, -6). Slope is (0 - (-6))/(4 - 0) = 6/4 = 1.5.

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52. 52Linear inequalitiesChallenge

    For how many integers x is -2 < 3x - 5 ≤ 7?For how many integers x is -2 < 3x - 5 ≤ 7?

    • A33
    • B55
    • C44
    • D22
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    Answer: 33

    Add 5 throughout: 3 < 3x ≤ 12. Divide by 3: 1 < x ≤ 4. The integers are 2, 3 and 4, so there are three.Add 5 throughout: 3 < 3x ≤ 12. Divide by 3: 1 < x ≤ 4. The integers are 2, 3 and 4, so there are three.

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53. 53Function notationStandard

    If f(x) = x² + 1, which value of x gives f(x) = f(-3)?If f(x) = x^2 + 1, which value of x gives f(x) = f(-3)?

    • A1010
    • B33
    • C99
    • D-9-9
    Show the answer and explanationHide the explanation

    Answer: 33

    f(-3) = 10, and f(3) = 10 as well because squaring removes the sign. The value 10 is the output, not the input.f(-3) = 10, and f(3) = 10 as well because squaring removes the sign. The value 10 is the output, not the input.

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54. 54Proportional relationshipsStandard

    On a map, 3 centimeters represents 12 kilometers. Enter the number of kilometers represented by 7.5 centimeters.On a map, 3 centimeters represents 12 kilometers. Enter the number of kilometers represented by 7.5 centimeters.

    Student-produced response: write your own answer rather than choosing one.

    Show the answer and explanationHide the explanation

    Answer: 3030

    The scale is 123 = 4 kilometers per centimeter, and 4 × 7.5 = 30 kilometers.The scale is 12/3 = 4 kilometers per centimeter, and 4 × 7.5 = 30 kilometers.

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55. 55Equivalent expressionsChallenge

    If a = 3b and b = 2c, which expression gives a in terms of c?If a = 3b and b = 2c, which expression gives a in terms of c?

    • A6c6c
    • Bc/6c/6
    • C1.5c1.5c
    • D5c5c
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    Answer: 6c6c

    Substitute b = 2c into a = 3b: a = 3(2c) = 6c. Adding the coefficients rather than multiplying gives 5c, which is the trap.Substitute b = 2c into a = 3b: a = 3(2c) = 6c. Adding the coefficients rather than multiplying gives 5c, which is the trap.

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56. 56Interpreting linear modelsStandard

    A tank drains according to V = 500 - 25t, where V is liters and t is minutes. How long does it take to empty?A tank drains according to V = 500 - 25t, where V is liters and t is minutes. How long does it take to empty?

    • A475 minutes475 minutes
    • B25 minutes25 minutes
    • C20 minutes20 minutes
    • D500 minutes500 minutes
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    Answer: 20 minutes20 minutes

    Empty means V = 0, so 25t = 500 and t = 20 minutes. The 25 is the rate, not the time.Empty means V = 0, so 25t = 500 and t = 20 minutes. The 25 is the rate, not the time.

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