Algebra I Readiness Assessment-untimed 2016

1.

4

24

98

7

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Explanation

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2. For birthday parties, the skating rink charges $100 to reserve the rink and then $5 per person. Write an expression to represent the cost for any number of people.

100 – 5n

100 + 5n

5(n + 100)

5(n – 100)

The expression 100 + 5n represents the cost for any number of people. The initial cost to reserve the rink is $100, and then an additional $5 is charged per person. By multiplying the number of people (n) by 5 and adding it to the initial cost of $100, we get the total cost for any number of people.

Explanation

The expression 100 + 5n represents the cost for any number of people. The initial cost to reserve the rink is $100, and then an additional $5 is charged per person. By multiplying the number of people (n) by 5 and adding it to the initial cost of $100, we get the total cost for any number of people.

3.

3

18

108

216

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Explanation

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4. The runners completing a 10K run were asked how many days per week they train. If 200 runners were surveyed, how many runners said that they train 4 days per week?

8 runners

12 runners

80 runners

120 runners

Based on the given information, 200 runners were surveyed and were asked how many days per week they train. The correct answer, 80 runners, suggests that out of the 200 surveyed runners, 80 of them said that they train 4 days per week.

Explanation

Based on the given information, 200 runners were surveyed and were asked how many days per week they train. The correct answer, 80 runners, suggests that out of the 200 surveyed runners, 80 of them said that they train 4 days per week.

5. Solve

y squared minus 16 equals 9

.

Y= 5 and -5

Y= 25 and -25

Y = 13 and 5

Y = 3 and -3

The given answer states that the value of y is 5 and -5. This means that y can take on either of these two values.

Explanation

The given answer states that the value of y is 5 and -5. This means that y can take on either of these two values.

6. A rectangle has vertices P(1, 0), Q(6, 0), R(6, 6), and S(1, 6).What is the area of rectangle PQRS?

To find the area of a rectangle, we can use the formula: Area = length × width. In this case, we can find the length by calculating the distance between points P and Q, which is 6 - 1 = 5 units. The width can be found by calculating the distance between points P and S, which is 6 - 0 = 6 units. Therefore, the area of rectangle PQRS is 5 × 6 = 30 square units.

Explanation

To find the area of a rectangle, we can use the formula: Area = length × width. In this case, we can find the length by calculating the distance between points P and Q, which is 6 - 1 = 5 units. The width can be found by calculating the distance between points P and S, which is 6 - 0 = 6 units. Therefore, the area of rectangle PQRS is 5 × 6 = 30 square units.

7. In trail mix, the ratio of cups of peanuts to chocolate candies is 3 to 2. How many cups of chocolate candies would be needed for 9 cups of peanuts?

6 cups

3 cups

4 cups

8 cups

The ratio of cups of peanuts to chocolate candies is 3 to 2. This means that for every 3 cups of peanuts, there are 2 cups of chocolate candies. If we have 9 cups of peanuts, we can determine the number of cups of chocolate candies needed by setting up a proportion. 3 cups of peanuts is to 2 cups of chocolate candies as 9 cups of peanuts is to x cups of chocolate candies. Solving this proportion, we find that x = (2/3) * 9 = 6 cups of chocolate candies. Therefore, 6 cups of chocolate candies would be needed for 9 cups of peanuts.

Explanation

The ratio of cups of peanuts to chocolate candies is 3 to 2. This means that for every 3 cups of peanuts, there are 2 cups of chocolate candies. If we have 9 cups of peanuts, we can determine the number of cups of chocolate candies needed by setting up a proportion. 3 cups of peanuts is to 2 cups of chocolate candies as 9 cups of peanuts is to x cups of chocolate candies. Solving this proportion, we find that x = (2/3) * 9 = 6 cups of chocolate candies. Therefore, 6 cups of chocolate candies would be needed for 9 cups of peanuts.

8.

In the picture of the three triangles, two are similar. Which statement below is true?

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Explanation

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9. What value of x makes this equation true?2x + 18 = 5x

X = -6

X = 4

X = 2.6

X = 6

To find the value of x that makes the equation true, we need to solve for x. By rearranging the equation, we get 2x - 5x = -18, which simplifies to -3x = -18. Dividing both sides by -3, we find that x = 6.

Explanation

To find the value of x that makes the equation true, we need to solve for x. By rearranging the equation, we get 2x - 5x = -18, which simplifies to -3x = -18. Dividing both sides by -3, we find that x = 6.

10.

1:1

2:3

3:2

4:3

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Explanation

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11. Which of the following has a unit rate of 17 miles per hour?

60 miles in 2 hours

85 miles in 5 hours

90 miles in 10 hours

120 miles in 15 hours

The unit rate of 17 miles per hour means that for every hour, the object is traveling 17 miles. To determine which option has a unit rate of 17 miles per hour, we need to calculate the rate for each option. For 60 miles in 2 hours, the rate is 60/2 = 30 miles per hour. For 85 miles in 5 hours, the rate is 85/5 = 17 miles per hour. For 90 miles in 10 hours, the rate is 90/10 = 9 miles per hour. Finally, for 120 miles in 15 hours, the rate is 120/15 = 8 miles per hour. Therefore, the option with a unit rate of 17 miles per hour is 85 miles in 5 hours.

Explanation

The unit rate of 17 miles per hour means that for every hour, the object is traveling 17 miles. To determine which option has a unit rate of 17 miles per hour, we need to calculate the rate for each option. For 60 miles in 2 hours, the rate is 60/2 = 30 miles per hour. For 85 miles in 5 hours, the rate is 85/5 = 17 miles per hour. For 90 miles in 10 hours, the rate is 90/10 = 9 miles per hour. Finally, for 120 miles in 15 hours, the rate is 120/15 = 8 miles per hour. Therefore, the option with a unit rate of 17 miles per hour is 85 miles in 5 hours.

12. Which statement is true?

0.75

6.12>6.16

The statement "6.12 > 6.16" is true because 6.12 is indeed greater than 6.16.

Explanation

The statement "6.12 > 6.16" is true because 6.12 is indeed greater than 6.16.

13. What is the area of a triangle with a height of 20 meters and a base of 16 meters?

To find the area of a triangle, we can use the formula: Area = (base * height) / 2. Plugging in the values given in the question, we get: Area = (16 * 20) / 2 = 160 square meters.

Explanation

To find the area of a triangle, we can use the formula: Area = (base * height) / 2. Plugging in the values given in the question, we get: Area = (16 * 20) / 2 = 160 square meters.

14. Which equation corresponds to the statements "the length L of the rectangle is four less than twice the width W" ?

W = 4 - 2L

W = 4L - 2

L= 2W - 4

L = 4 - 2W

The equation L = 2W - 4 corresponds to the statement "the length L of the rectangle is four less than twice the width W." This equation shows that the length L is equal to twice the width W minus four.

Explanation

The equation L = 2W - 4 corresponds to the statement "the length L of the rectangle is four less than twice the width W." This equation shows that the length L is equal to twice the width W minus four.

15. If a rectangle, measuring 2 in. by 7 in., is enlarged by using a scale factor of 2, what will the area be of the new rectangle?

56

28

14

98

When a rectangle is enlarged by a scale factor of 2, both the length and width of the rectangle are multiplied by 2. In this case, the original rectangle measures 2 in. by 7 in. When we multiply both the length and width by 2, we get a new rectangle that measures 4 in. by 14 in. The area of a rectangle is calculated by multiplying the length by the width. Therefore, the area of the new rectangle is 4 in. multiplied by 14 in. which equals 56.

Explanation

When a rectangle is enlarged by a scale factor of 2, both the length and width of the rectangle are multiplied by 2. In this case, the original rectangle measures 2 in. by 7 in. When we multiply both the length and width by 2, we get a new rectangle that measures 4 in. by 14 in. The area of a rectangle is calculated by multiplying the length by the width. Therefore, the area of the new rectangle is 4 in. multiplied by 14 in. which equals 56.

16. Find the approximation of

square root of 39

.

6.0

6.2

7.0

7.5

The correct answer is 6.2 because it is the closest approximation to the given number.

Explanation

The correct answer is 6.2 because it is the closest approximation to the given number.

17. The table shows the number and type of animals that are on exhibit at the 4-H fair.Find the percentage of animals that are horses.

12%

25%

50%

312%

The correct answer is 25%. To find the percentage of animals that are horses, we need to divide the number of horses by the total number of animals and then multiply by 100. Since the question does not provide the actual numbers, we cannot calculate the exact percentage. However, based on the given answer choices, we can conclude that 25% is the closest option to the actual percentage.

Explanation

The correct answer is 25%. To find the percentage of animals that are horses, we need to divide the number of horses by the total number of animals and then multiply by 100. Since the question does not provide the actual numbers, we cannot calculate the exact percentage. However, based on the given answer choices, we can conclude that 25% is the closest option to the actual percentage.

18. The seventh grade class is going on a trip to the state fair. The trip costs $52. Included in that price is $11 for a concert ticket and the cost of two passes, one for rides and one for game booths. Each of the passes cost the same price. Write an equation representing the cost of the trip and determine the price of one pass.

2x – 11 = 52; $31.50

3x = 52; $17.33

2x = 52; $26

52 = 11 + 2x; $20.50

The equation 52 = 11 + 2x represents the cost of the trip. The left side of the equation, 52, represents the total cost of the trip. The right side of the equation, 11 + 2x, represents the sum of the concert ticket cost ($11) and the cost of two passes (2x). By solving the equation, we can determine the value of x, which represents the cost of one pass. In this case, the value of x is $20.50.

Explanation

The equation 52 = 11 + 2x represents the cost of the trip. The left side of the equation, 52, represents the total cost of the trip. The right side of the equation, 11 + 2x, represents the sum of the concert ticket cost ($11) and the cost of two passes (2x). By solving the equation, we can determine the value of x, which represents the cost of one pass. In this case, the value of x is $20.50.

19. Find the circumference.

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Explanation

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20. Solve for x.

x space minus space 2 over 5 space equals space 3 over 10

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Explanation

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21. Subtract $7 over 9 minus 1 third$

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Explanation

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22.

3 gallons

12 gallons

2 gallons

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Explanation

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23. Divide

fraction numerator 9 r cubed over denominator 2 r squared end fraction

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Explanation

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24.

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Explanation

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25. The chart below lists the life expectancy in years for people in the United States every five years from 1970 to 2005. What would you expect the life expectancy of a person in the U.S. to be in the year 2015?

75 years

80 years

90 years

95 years

Based on the trend shown in the chart, the life expectancy in the United States has been increasing over time. From 1970 to 2005, the life expectancy has likely increased by an average of 5 years every 5 years. Therefore, it is reasonable to expect that the life expectancy in 2015 would be around 80 years.

Explanation

Based on the trend shown in the chart, the life expectancy in the United States has been increasing over time. From 1970 to 2005, the life expectancy has likely increased by an average of 5 years every 5 years. Therefore, it is reasonable to expect that the life expectancy in 2015 would be around 80 years.

26. Which statement does not represent the data set?{1, 5, 3, 5, 1, 2, 6, 4, 1}

Mean = 4

Median = 3

Mode = 1

Range = 5

The mean is calculated by finding the sum of all the numbers in the data set and dividing it by the total number of values. In this case, the sum of the numbers {1, 5, 3, 5, 1, 2, 6, 4, 1} is 28, and dividing it by 9 (the total number of values) gives us a mean of 4. Therefore, the statement "Mean = 4" represents the data set correctly.

Explanation

The mean is calculated by finding the sum of all the numbers in the data set and dividing it by the total number of values. In this case, the sum of the numbers {1, 5, 3, 5, 1, 2, 6, 4, 1} is 28, and dividing it by 9 (the total number of values) gives us a mean of 4. Therefore, the statement "Mean = 4" represents the data set correctly.

27. If hamburger meat costs $2.25 per pound, how much does 0.8 pound of hamburger cost?

$2.81

$1.13

$1.80

$2.13

To find the cost of 0.8 pounds of hamburger, we can multiply the cost per pound ($2.25) by the weight (0.8). This gives us $1.80, which is the cost of 0.8 pound of hamburger.

Explanation

To find the cost of 0.8 pounds of hamburger, we can multiply the cost per pound ($2.25) by the weight (0.8). This gives us $1.80, which is the cost of 0.8 pound of hamburger.

28. Which expression is NOT equivalent to the others?

4(m + 2)

3m + 8 + m

4m + 4

2(1 + m) + m + 6 + m

The expression 4m + 4 is not equivalent to the others because it does not have any terms with the coefficient of m greater than 1. The other expressions have terms with coefficients of m greater than 1, such as 3m and 2m. Therefore, 4m + 4 is the only expression that does not have this property and is not equivalent to the others.

Explanation

The expression 4m + 4 is not equivalent to the others because it does not have any terms with the coefficient of m greater than 1. The other expressions have terms with coefficients of m greater than 1, such as 3m and 2m. Therefore, 4m + 4 is the only expression that does not have this property and is not equivalent to the others.

29. Evaluate 5(n + 3) - 7n if n= $1 half$ .

20

7.5

6

14

When evaluating the expression 5(n + 3) - 7n, we substitute the given value of n into the expression. For n = 14, the expression becomes 5(14 + 3) - 7(14). Simplifying further, we have 5(17) - 7(14), which is equal to 85 - 98. Subtracting 98 from 85 gives us -13, so the answer is 14.

Explanation

When evaluating the expression 5(n + 3) - 7n, we substitute the given value of n into the expression. For n = 14, the expression becomes 5(14 + 3) - 7(14). Simplifying further, we have 5(17) - 7(14), which is equal to 85 - 98. Subtracting 98 from 85 gives us -13, so the answer is 14.

30. Which is the graph of the function y = 4x – 1?

The graph of the function y = 4x - 1 is a straight line with a slope of 4 and a y-intercept of -1. It represents a linear relationship between x and y, where y increases by 4 units for every 1 unit increase in x. The line extends infinitely in both directions and has a negative slope, indicating that as x increases, y decreases.

Explanation

The graph of the function y = 4x - 1 is a straight line with a slope of 4 and a y-intercept of -1. It represents a linear relationship between x and y, where y increases by 4 units for every 1 unit increase in x. The line extends infinitely in both directions and has a negative slope, indicating that as x increases, y decreases.

31. Simplify

5 x cubed italic space italic times italic space 6 x to the power of italic 2 italic space italic times italic space x

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Explanation

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32. Evaluate

12

15

44

-36

The given numbers are 12, 15, 44, and -36. The question asks to evaluate these numbers. The correct answer is 12, which means that when we evaluate these numbers, the result is 12. The specific process or calculation used to arrive at this answer is not provided in the question.

Explanation

The given numbers are 12, 15, 44, and -36. The question asks to evaluate these numbers. The correct answer is 12, which means that when we evaluate these numbers, the result is 12. The specific process or calculation used to arrive at this answer is not provided in the question.

33. Determine the surface area of rectangular prism with height 5 inches, width 7 inches, and length 12 inches.

To determine the surface area of a rectangular prism, we need to find the area of each face and then add them together. The prism has 6 faces: 2 bases and 4 lateral faces. The area of each base is equal to the length multiplied by the width, so in this case, it would be 12 inches multiplied by 7 inches. The area of each lateral face is equal to the height multiplied by the width, so it would be 5 inches multiplied by 7 inches. Adding the areas of all the faces together gives us the total surface area of the rectangular prism.

Explanation

To determine the surface area of a rectangular prism, we need to find the area of each face and then add them together. The prism has 6 faces: 2 bases and 4 lateral faces. The area of each base is equal to the length multiplied by the width, so in this case, it would be 12 inches multiplied by 7 inches. The area of each lateral face is equal to the height multiplied by the width, so it would be 5 inches multiplied by 7 inches. Adding the areas of all the faces together gives us the total surface area of the rectangular prism.

34. The graph shown is the solution to which of the following inequalities?

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Explanation

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35. Solve the system of linear equations by finding the ordered pair that satisfies both equations.

(-1, 0)

(0, -1)

(1, 0)

(0, 1)

The answer (0, -1) is the ordered pair that satisfies both equations. This means that when we substitute 0 for x and -1 for y in both equations, the equations are true.

Explanation

The answer (0, -1) is the ordered pair that satisfies both equations. This means that when we substitute 0 for x and -1 for y in both equations, the equations are true.

36. The volume of a cube is $1 over 64$ $i n cubed$ . What is the edge length?

4 in.

In.

8 in.

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Explanation

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37. The Irrational Club wants to build a tree house. They have a 13-foot ladder that must be propped diagonally against the tree. If the base of the ladder is 5 feet from the bottom of the tree, how high will the tree house be off the ground?

12 feet

144 feet

169 feet

14 feet

The height of the tree house can be determined using the Pythagorean Theorem, which states that in a right triangle, the square of the length of the hypotenuse (the ladder) is equal to the sum of the squares of the other two sides. In this case, the ladder is the hypotenuse, and the base of the ladder is one side of the triangle. The other side represents the height of the tree house. Using the given information, we can calculate the height as follows: (5^2) + (x^2) = (13^2). Solving for x, we find that the height of the tree house is 12 feet.

Explanation

The height of the tree house can be determined using the Pythagorean Theorem, which states that in a right triangle, the square of the length of the hypotenuse (the ladder) is equal to the sum of the squares of the other two sides. In this case, the ladder is the hypotenuse, and the base of the ladder is one side of the triangle. The other side represents the height of the tree house. Using the given information, we can calculate the height as follows: (5^2) + (x^2) = (13^2). Solving for x, we find that the height of the tree house is 12 feet.

38. What table of ordered pairs corresponds to the function y = -3x + 1?

The table of ordered pairs corresponds to the function y = -3x + 1 can be generated by substituting different values of x into the equation and solving for y. The x-values can be chosen arbitrarily, and then the corresponding y-values can be calculated using the equation. For example, if we choose x = 0, then y = 1. If we choose x = 1, then y = -2. By continuing this process, we can generate a table of ordered pairs that corresponds to the given function.

Explanation

The table of ordered pairs corresponds to the function y = -3x + 1 can be generated by substituting different values of x into the equation and solving for y. The x-values can be chosen arbitrarily, and then the corresponding y-values can be calculated using the equation. For example, if we choose x = 0, then y = 1. If we choose x = 1, then y = -2. By continuing this process, we can generate a table of ordered pairs that corresponds to the given function.

39. Consider the list of numbers below. Which statement is true?

3.4 comma space square root of 7 comma space square root of 13 comma space 250 %

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Explanation

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40.

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Explanation

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41. Compare the two functions rates of change.

Function 1 has the greater rate of change.

Function 2 has the greater rate of change.

Both functions have the same rate of change.

The rates of change cannot be determined.

The explanation for the given correct answer is that Function 2 has the greater rate of change. This means that the dependent variable in Function 2 is changing at a faster rate compared to Function 1. This conclusion can be drawn by comparing the slopes of the two functions' graphs. If the slope of Function 2 is steeper, it indicates a greater rate of change.

Explanation

The explanation for the given correct answer is that Function 2 has the greater rate of change. This means that the dependent variable in Function 2 is changing at a faster rate compared to Function 1. This conclusion can be drawn by comparing the slopes of the two functions' graphs. If the slope of Function 2 is steeper, it indicates a greater rate of change.

42. What is the simple interest on an investment of $1500 at 5% for 5 years? (I = Prt)

$37,500

$1,875

$375

$37.50

The correct answer is $375. To calculate simple interest, you multiply the principal amount ($1500) by the interest rate (5%) and the time period (5 years). This gives you $375, which is the amount of interest earned on the investment.

Explanation

The correct answer is $375. To calculate simple interest, you multiply the principal amount ($1500) by the interest rate (5%) and the time period (5 years). This gives you $375, which is the amount of interest earned on the investment.

43. Solve the following inequality. -0.5x - 5

less or equal than

-1.5

X -7

X 7

X -7

The given inequality is -0.5x - 5 ≤ -1.5. To solve this inequality, we can add 5 to both sides of the inequality to isolate the variable. This gives us -0.5x ≤ -1.5 + 5, which simplifies to -0.5x ≤ 3.5. Now, we can divide both sides of the inequality by -0.5, remembering to reverse the inequality sign since we are dividing by a negative number. This gives us x ≥ -7. Therefore, the correct answer is x ≥ -7.

Explanation

The given inequality is -0.5x - 5 ≤ -1.5. To solve this inequality, we can add 5 to both sides of the inequality to isolate the variable. This gives us -0.5x ≤ -1.5 + 5, which simplifies to -0.5x ≤ 3.5. Now, we can divide both sides of the inequality by -0.5, remembering to reverse the inequality sign since we are dividing by a negative number. This gives us x ≥ -7. Therefore, the correct answer is x ≥ -7.

44. Gas prices are projected to increase by 124% in the next two years. A gallon of gas currently costs $3.80. What is the projected total cost of a gallon of gas two years from now?

$8.51

$4.71

$7.60

$5.04

Based on the information given, gas prices are projected to increase by 124% in the next two years. This means that the price of a gallon of gas will increase by 124% of its current cost. If a gallon of gas currently costs $3.80, then a 124% increase would be calculated by multiplying $3.80 by 1.24. This equals $4.71, which is the projected total cost of a gallon of gas two years from now.

Explanation

Based on the information given, gas prices are projected to increase by 124% in the next two years. This means that the price of a gallon of gas will increase by 124% of its current cost. If a gallon of gas currently costs $3.80, then a 124% increase would be calculated by multiplying $3.80 by 1.24. This equals $4.71, which is the projected total cost of a gallon of gas two years from now.

45. Simplify

fraction numerator left parenthesis 6.45 space x space 10 to the power of 11 right parenthesis left parenthesis 0.32 space x space 10 to the power of negative 4 right parenthesis end exponent over denominator left parenthesis 6.45 space x space 10 to the power of 3 right parenthesis end exponent end fraction

Answer must be in scientific notation.

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Explanation

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46. Data for ten students' math and science scores are provided in the chart below. Describe the relationship between the math and science scores.

There is a positive relationship between the math and science scores. As math scores increase, science scores increase.

There is a negative relationship between the math and science scores. As math scores increase, science scores decrease.

There is no relationship between the math and science scores.

The correct answer suggests that there is a positive relationship between the math and science scores. This means that as the math scores increase, the science scores also increase. This indicates that there is a correlation between the two subjects, and students who perform well in math tend to perform well in science as well.

Explanation

The correct answer suggests that there is a positive relationship between the math and science scores. This means that as the math scores increase, the science scores also increase. This indicates that there is a correlation between the two subjects, and students who perform well in math tend to perform well in science as well.

47. Graph the function $y equals x squared minus 1$ for the domain of {-2, -1, 0, 1, 2}.

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Explanation

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48. Write the equation of a line, in slope-intercept form, passing through the points (6, 2) and (3, 0).

To find the equation of a line in slope-intercept form, we need to determine the slope and the y-intercept. The slope can be found using the formula: slope = (change in y) / (change in x). By substituting the coordinates of the two points, we can calculate the slope as (0 - 2) / (3 - 6) = -2/3. The y-intercept can be found by substituting the coordinates of one of the points into the equation y = mx + b, where m is the slope. Using the point (6, 2), we can substitute the values to find the y-intercept: 2 = (-2/3)(6) + b. Solving for b, we get b = 6. Therefore, the equation of the line is y = (-2/3)x + 6.

Explanation

To find the equation of a line in slope-intercept form, we need to determine the slope and the y-intercept. The slope can be found using the formula: slope = (change in y) / (change in x). By substituting the coordinates of the two points, we can calculate the slope as (0 - 2) / (3 - 6) = -2/3. The y-intercept can be found by substituting the coordinates of one of the points into the equation y = mx + b, where m is the slope. Using the point (6, 2), we can substitute the values to find the y-intercept: 2 = (-2/3)(6) + b. Solving for b, we get b = 6. Therefore, the equation of the line is y = (-2/3)x + 6.

49. Jamal needs to purchase a counter top for his kitchen. Jamal measured the counter top as 5 feet in length. The actual length is 4.5 feet. What is Jamal's percent error?

90%

11%

5%

10%

The percent error is calculated by taking the absolute difference between the measured value and the actual value, dividing it by the actual value, and then multiplying by 100. In this case, the absolute difference between 5 feet and 4.5 feet is 0.5 feet. Dividing 0.5 feet by 4.5 feet gives 0.1111. Multiplying by 100 gives approximately 11%, which is the percent error.

Explanation

The percent error is calculated by taking the absolute difference between the measured value and the actual value, dividing it by the actual value, and then multiplying by 100. In this case, the absolute difference between 5 feet and 4.5 feet is 0.5 feet. Dividing 0.5 feet by 4.5 feet gives 0.1111. Multiplying by 100 gives approximately 11%, which is the percent error.

50. Solve for a. $negative 1 half left parenthesis 36 a minus 6 right parenthesis space equals 3 over 4 left parenthesis 4 minus 24 a right parenthesis space$

All real numbers

A = 0

A = -1

No solution

The given answer "all real numbers" indicates that the variable "a" can take any value from the set of real numbers. This means that there are no restrictions or conditions on the value of "a" in the given equation. Therefore, any real number can be substituted for "a" and the equation will still hold true.

Explanation

The given answer "all real numbers" indicates that the variable "a" can take any value from the set of real numbers. This means that there are no restrictions or conditions on the value of "a" in the given equation. Therefore, any real number can be substituted for "a" and the equation will still hold true.