Free practice questions/GRE General Test

GRE General Test — Algebra

36 free practice questions with full explanations.

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Question 1

If g(x) = 2x^2 - x + 4, what is g(-1)?

A) 3
B) 5
C) 7
D) 9
E) 11

Show answer & explanation

Correct answer: C) 7

g(-1) = 2(-1)^2 - (-1) + 4 = 2(1) + 1 + 4 = 2 + 1 + 4 = 7.

Question 2

If x^2 = 64 and x < 0, what is the value of x - 3?

A) -11
B) -5
C) 5
D) 8
E) 11

Show answer & explanation

Correct answer: A) -11

Since x^2 = 64 and x < 0, x = -8. x - 3 = -8 - 3 = -11.

Question 3

Which of the following describes all values of x satisfying 3x - 5 > 10?

A) x > 5
B) x < 5
C) x > 15
D) x > 3
E) x < 15

Show answer & explanation

Correct answer: A) x > 5

3x - 5 > 10. 3x > 15. x > 5.

Question 4

If 4x - 9 = 2x + 11, what is the value of x?

A) 5
B) 8
C) 10
D) 12
E) 15

Show answer & explanation

Correct answer: C) 10

4x - 9 = 2x + 11. 2x = 20. x = 10.

Question 5

A line has slope 3 and passes through the point (2, -1). What is the y-intercept of this line?

A) -7
B) -5
C) 3
D) 5
E) 7

Show answer & explanation

Correct answer: A) -7

y - (-1) = 3(x - 2) -> y + 1 = 3x - 6 -> y = 3x - 7. The y-intercept (x=0) is -7.

Question 6

If f(x) = 2x^2 - 5, what is f(4)?

A) 27
B) 11
C) 32
D) 3
E) 21

Show answer & explanation

Correct answer: A) 27

f(4)=2(16)-5=32-5=27.

Question 7

If 4x - 9 = 2x + 7, what is x?

A) 10
B) 8
C) 6
D) 7
E) 9

Show answer & explanation

Correct answer: B) 8

2x=16. x=8.

Question 8

If 3^x = 81, what is x?

A) 9
B) 5
C) 4
D) 3
E) 27

Show answer & explanation

Correct answer: C) 4

81=3^4. x=4.

Question 9

Factor: x^2 - 49

A) (x-7)(x-7)
B) (x+49)(x-1)
C) (x+7)(x+7)
D) (x-7)(x+7)
E) (x-49)(x+1)

Show answer & explanation

Correct answer: D) (x-7)(x+7)

This is a difference of squares: x^2-49=(x-7)(x+7).

Question 10

If x + y = 15 and x - y = 5, what is y?

A) 10
B) 7.5
C) 3
D) 4
E) 5

Show answer & explanation

Correct answer: E) 5

Subtracting: 2y=10. y=5.

Question 11

If x^2 - 5x - 14 = 0, what are the possible values of x?

A) 2 and -7
B) -2 and 7
C) 7 and -2
D) 14 and -1
E) -14 and 1

Show answer & explanation

Correct answer: C) 7 and -2

Factor: x^2 - 5x - 14 = (x-7)(x+2) = 0. So x = 7 or x = -2.

Question 12

If f(x) = 2x^2 - 3x + 1, what is f(-2)?

A) 3
B) 7
C) 11
D) 15
E) 19

Show answer & explanation

Correct answer: D) 15

f(-2) = 2(-2)^2 - 3(-2) + 1 = 2(4) + 6 + 1 = 8 + 6 + 1 = 15.

Question 13

If 2^(x+3) = 32, what is the value of x?

A) 1
B) 2
C) 3
D) 4
E) 5

Show answer & explanation

Correct answer: B) 2

32 = 2^5. So x + 3 = 5, meaning x = 2.

Question 14

For what value of k does the equation 4x + k = 2(x + 5) have infinitely many solutions?

A) 5
B) 8
C) 10
D) 12
E) There is no such value

Show answer & explanation

Correct answer: E) There is no such value

4x + k = 2x + 10 simplifies to 2x + k = 10, or x = (10-k)/2, which is a single linear equation in x -- it has exactly one solution for any k (unless the x terms cancel, which they don't here since 4 ≠ 2), never infinitely many.

Question 15

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

A) 2
B) 3
C) 4
D) 5
E) 6

Show answer & explanation

Correct answer: C) 4

From the second equation, x = y + 2. Substitute into the first: 3(y+2) + 2y = 16 -> 3y + 6 + 2y = 16 -> 5y = 10 -> y = 2. Then x = y + 2 = 4.

Question 16

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

A) 1
B) 5
C) 15
D) 21
E) 30

Show answer & explanation

Correct answer: C) 15

3x - 7 = 2x + 8. Subtract 2x from both sides: x - 7 = 8. Add 7: x = 15.

Question 17

If f(x) = x^2 + 2x - 3, what is f(3)?

A) 6
B) 9
C) 12
D) 15
E) 18

Show answer & explanation

Correct answer: C) 12

f(3) = 3^2 + 2(3) - 3 = 9 + 6 - 3 = 12.

Question 18

If (x - 4)(x + 6) = 0 and x > 0, what is the value of x?

A) -6
B) -4
C) 4
D) 6
E) 24

Show answer & explanation

Correct answer: C) 4

The equation is satisfied when x = 4 or x = -6. Since x > 0, x = 4.

Question 19

If 5x + 3 = 3x + 15, what is the value of x?

A) 3
B) 4
C) 5
D) 6
E) 9

Show answer & explanation

Correct answer: D) 6

5x + 3 = 3x + 15. 2x = 12. x = 6.

Question 20

If x^2 - 3x - 10 = 0, what are the possible values of x?

A) 5 and -2
B) -5 and 2
C) 2 and -5
D) 10 and -1
E) -10 and 1

Show answer & explanation

Correct answer: A) 5 and -2

Factor: x^2 - 3x - 10 = (x-5)(x+2) = 0. So x = 5 or x = -2.

Question 21

If 3x + y = 20 and x = 3y, what is the value of y?

A) 2
B) 3
C) 4
D) 5
E) 6

Show answer & explanation

Correct answer: A) 2

Substitute x = 3y into the first equation: 3(3y) + y = 20. 9y + y = 10y = 20. y = 2. (Then x = 3y = 6, and checking: 3(6) + 2 = 20, correct.)

Question 22

If h(x) = 3x^2 + x - 2, what is h(2)?

A) 8
B) 10
C) 12
D) 14
E) 16

Show answer & explanation

Correct answer: C) 12

h(2) = 3(2)^2 + 2 - 2 = 3(4) + 2 - 2 = 12 + 2 - 2 = 12.

Question 23

If 6x + 4 = 3x + 22, what is the value of x?

A) 4
B) 5
C) 6
D) 7
E) 8

Show answer & explanation

Correct answer: C) 6

6x + 4 = 3x + 22. 3x = 18. x = 6.

Question 24

A line passes through points (0, 3) and (4, 11). What is the slope of this line?

A) 1
B) 2
C) 3
D) 4
E) 8

Show answer & explanation

Correct answer: B) 2

Slope = (11-3)/(4-0) = 8/4 = 2.

Question 25

Which of the following describes all values of x satisfying 4x + 7 <= 23?

A) x <= 4
B) x <= 8
C) x >= 4
D) x <= 16
E) x >= 8

Show answer & explanation

Correct answer: A) x <= 4

4x + 7 <= 23. 4x <= 16. x <= 4.

Question 26

If x^2 - 2x - 15 = 0, what are the possible values of x?

A) 5 and -3
B) -5 and 3
C) 3 and -5
D) 15 and -1
E) -15 and 1

Show answer & explanation

Correct answer: A) 5 and -3

Factor: x^2 - 2x - 15 = (x-5)(x+3) = 0. So x = 5 or x = -3.

Question 27

If x/y = 3/4 and y/z = 2/5, what is x/z?

A) 6/9
B) 2/3
C) 8/15
D) 5/12
E) 3/10

Show answer & explanation

Correct answer: E) 3/10

x/z = (x/y)(y/z) = (3/4)(2/5) = 6/20 = 3/10.

Question 28

The sum of two consecutive even integers is 74. What is the larger integer?

A) 38
B) 36
C) 40
D) 37
E) 39

Show answer & explanation

Correct answer: A) 38

Let integers be n and n+2. n + (n+2) = 74 -> 2n = 72 -> n = 36. Larger = 38.

Question 29

If 2(x - 3) = 3(x + 1) - 5, what is the value of x?

A) 4
B) -4
C) -2
D) 2
E) -6

Show answer & explanation

Correct answer: B) -4

2x - 6 = 3x + 3 - 5 -> 2x - 6 = 3x - 2 -> -6 + 2 = 3x - 2x -> -4 = x.

Question 30

If f(x) = x^2 - 4x + 3, for what value(s) of x does f(x) = 0?

A) x = -1 or x = -3
B) x = 1 or x = -3
C) x = 1 or x = 3
D) x = 2 only
E) x = 0 or x = 4

Show answer & explanation

Correct answer: C) x = 1 or x = 3

x^2 - 4x + 3 = (x-1)(x-3) = 0, so x = 1 or x = 3.

Question 31

If 3^(x+1) = 81, what is the value of x?

A) 4
B) 2
C) 5
D) 3
E) 81

Show answer & explanation

Correct answer: D) 3

81 = 3^4, so x + 1 = 4, meaning x = 3.

Question 32

If x + y = 20 and x - y = 6, what is x?

A) 10
B) 13
C) 11
D) 12
E) 14

Show answer & explanation

Correct answer: B) 13

Adding the equations: 2x=26. x=13.

Question 33

If f(x) = 3x^2 - 2x + 1, what is f(3)?

A) 16
B) 18
C) 20
D) 22
E) 24

Show answer & explanation

Correct answer: D) 22

f(3)=3(9)-2(3)+1=27-6+1=22.

Question 34

If 6x - 5 = 4x + 13, what is x?

A) 9
B) 6
C) 7
D) 8
E) 10

Show answer & explanation

Correct answer: A) 9

6x-5=4x+13. 2x=18. x=9.

Question 35

If 2^x = 128, what is x?

A) 5
B) 6
C) 8
D) 9
E) 7

Show answer & explanation

Correct answer: E) 7

128=2^7. x=7.

Question 36

Factor: x^2 - 81

A) (x-9)(x-9)
B) (x+81)(x-1)
C) (x-9)(x+9)
D) (x+9)(x+9)
E) (x-81)(x+1)

Show answer & explanation

Correct answer: C) (x-9)(x+9)

This is a difference of squares: x^2-81=(x-9)(x+9).

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