GRE Math : Algebra

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Example Questions

Example Question #91 : Expressions

x^2-4x=-3

Quantity A: $\frac{1}{x}$

Quantity B: $\frac{1}{4}$

Possible Answers:

The two quantities are equal.

The relationship cannot be determined.

Quantity A is larger.

Quantity B is larger.

Correct answer:

Quantity A is larger.

Explanation:

Since there is an x^2 term in the equation

x^2-4x=-3

We must acknowledge the possibility of multiple values that satisfy the equation.

Rewrite it so that it is set equal to zero:

x^2-4x+3=0

For an equation of the form

ax^2+bx+c=0

The solutions for can be found using the quadratic formula:

$x_{1,2}=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$

$x_{1,2}=\frac{-(-4) \pm \sqrt{(-4)^2-4(1)(3)}}{2(1)}=\frac{4 \pm \sqrt{4}}{2}=1,3$

The two possible values for Quantity A are

$\frac{1}{1},\frac{1}{3}$

Since $1>\frac{1}{4};\frac{1}{3}>\frac{1}{4}$ , Quantity A must be larger.

$\frac{1}{3}=\frac{4}{12}; \frac{1}{4}=\frac{3}{12}$

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Example Question #92 : Expressions

The expression x?y is equivalent to $(x-y)^{|y-x|}$ . Evaluate 2?5

Possible Answers:

$-\frac{1}{27}$

$\frac{1}{27}$

$-\frac{1}{9}$

Correct answer:

Explanation:

Since x?y is equivalent to $(x-y)^{|y-x|}$ , we can rewrite 2?5 as

$(2-5)^{|5-2|}=-3^{|3|}=-3^3$

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Example Question #53 : How To Evaluate Algebraic Expressions

The expression x?y is equivalent to $\frac{y(x^2+1)}{x-y}$ . Evaluate the expression -9?2?4 .

Possible Answers:

$\frac{-820}{19}$

$\frac{820}{19}$

$\frac{800}{19}$

Correct answer:

Explanation:

To approach this problem, do it in parts, starting at the right of the equation.

If expression x?y is equivalent to $\frac{y(x^2+1)}{x-y}$ , then

$2?4=\frac{4(2^2+1)}{2-4}=\frac{4(5)}{-2}=-10$

-9?2?4 then becomes -9?-10

$\frac{-10(-9^2+1)}{-9-(-10)}=\frac{-10(82)}{1}=-820$

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Example Question #94 : Expressions

x^2+x=30

x>0

Quantity A:

Quantity B:

Possible Answers:

The relationship cannot be determined.

The two quantities are equal.

Quantity A is greater.

Quantity B is greater.

Correct answer:

The two quantities are equal.

Explanation:

Since there is an x^2 term in the equation

x^2+x=30

We must acknowledge the possibility of multiple values that satisfy the equation.

Rewrite it so that it is set equal to zero:

x^2+x-30=0

For an equation of the form

ax^2+bx+c=0

The solutions for can be found using the quadratic formula:

$x_{1,2}=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$

$x_{1,2}=\frac{-1 \pm \sqrt{1^2-4(1)(-30)}}{2(1)}=\frac{-1 \pm \sqrt{121}}{2}=-6,5$

Since x>0 , it must be that x=5

The two quantities are equal.

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Example Question #91 : Expressions

x^2+x=42

x<0

Quantity A: |x|

Quantity B:

Possible Answers:

The relationship cannot be determined.

Quantity B is greater.

Quantity A is greater.

The two quantities are equal.

Correct answer:

Quantity A is greater.

Explanation:

Since there is an x^2 term in the equation

x^2+x=42

We must acknowledge the possibility of multiple values that satisfy the equation.

Rewrite it so that it is set equal to zero:

x^2+x-42=0

For an equation of the form

ax^2+bx+c=0

The solutions for can be found using the quadratic formula:

$x_{1,2}=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$

$x_{1,2}=\frac{-1 \pm \sqrt{1^2-4(1)(-42)}}{2(1)}=\frac{-1 \pm \sqrt{169}}{2}=-7,6$

Since x<0 , it must be that x=-7

|-7|=7, 7>6

Quantity A is greater.

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Example Question #93 : Expressions

If x=4z and y=8q , all of the of following are equal to 3(x+y) except?

Possible Answers:

$36\left(\frac{1}{3}z+\frac{2}{9}q\right)$

$\frac{1}{3}(36z+72q)$

$24\left(\frac{1}{2}z+q\right)$

6(2z+4q)

12(z+2q)

Correct answer:

$36\left(\frac{1}{3}z+\frac{2}{9}q\right)$

Explanation:

A way to approach this problem is to realize that the coefficient for should be twice whatever coefficient is multiplied by in the parenthesis, and to look for a case where that does not hold true.

In the case of

$36\left(\frac{1}{3}z+\frac{2}{9}q\right)$

The coffecient for is $\frac{2}{3}$ times the coefficient for , rather than twice it, and so there is no way this could satisfy the condition of 3(4z+8q) .

This method of evaluation saves the trouble of multiplying each function out to check for equivalency.

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Example Question #92 : Expressions

x^2+12x=-32

Quantity A:

Quantity B: x^3

Possible Answers:

Quantity A is greater.

The relationship cannot be determined.

The two quantities are equal.

Quantity B is greater.

Correct answer:

Quantity A is greater.

Explanation:

Since there is an x^2 term in the equation

x^2+12x=-32

We must acknowledge the possibility of multiple values that satisfy the equation.

Rewrite it so that it is set equal to zero:

x^2+12x+32=0

For an equation of the form

ax^2+bx+c=0

The solutions for can be found using the quadratic formula:

$x_{1,2}=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$

$x_{1,2}=\frac{-12 \pm \sqrt{12^2-4(1)(32)}}{2(1)}=\frac{-12 \pm \sqrt{16}}{2}=-4,-8$

Now there are two potential values for , both of which are negative. Consider our quantities. For both of these values, Quantity A is greater.

x=-4;x^3=-64

x=-8;x^3=-512

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Example Question #93 : Expressions

2x^2+7x=4

Quantity A:

Quantity B: x^3

Possible Answers:

Quantity B is greater.

The relationship cannot be determined.

Quantity A is greater.

The two quantities are equal.

Correct answer:

Quantity A is greater.

Explanation:

Since there is an x^2 term in the equation

2x^2+7x=4

We must acknowledge the possibility of multiple values that satisfy the equation.

Rewrite it so that it is set equal to zero:

2x^2+7x-4=0

For an equation of the form

ax^2+bx+c=0

The solutions for can be found using the quadratic formula:

$x_{1,2}=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$

$x_{1,2}=\frac{-7 \pm \sqrt{7^2-4(2)(-4)}}{2(2)}=\frac{-7 \pm \sqrt{81}}{4}=-4,\frac{1}{2}$

So there are two possible values for , one negative and one positive. Let's consider our quantities for these two possibilities.
x=-4;x^3=-64;A>B

$x=\frac{1}{2};x^3=\frac{1}{8};A>B$

Even though we have a positive value for one of the values of , since it is less than one, the cube of it is smaller than it is.

Quantity A is greater.

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Example Question #94 : Expressions

3x^2-28x=-9

Quantity A:

Quantity B: x^2

Possible Answers:

Quantity B is greater.

Quantity A is greater.

The relationship cannot be determined.

The two quantities are equal.

Correct answer:

The relationship cannot be determined.

Explanation:

Since there is an x^2 term in the equation

3x^2-28x=-9

We must acknowledge the possibility of multiple values that satisfy the equation.

Rewrite it so that it is set equal to zero:

3x^2-28x+9=0

For an equation of the form

ax^2+bx+c=0

The solutions for can be found using the quadratic formula:

$x_{1,2}=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$

$x_{1,2}=\frac{-(-28) \pm \sqrt{(-28)^2-4(3)(9)}}{2(3)}=\frac{28 \pm \sqrt{676}}{6}=\frac{28\pm 26}{6}=\frac{1}{3},9$

So there are two possible values of , both of which are positive. Compare the two quantities for each potential value:

$x=\frac{1}{3};x^2=\frac{1}{9};A>B$

x=9;x^2=81;B>A

Depending on the value of , Quantity A or B could be greater. The relationship cannot be determined.

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Example Question #95 : Expressions

10x^2-7x=-1

Quantity A:

Quantity B: x^2

Possible Answers:

The two quantities are equal.

The relationship cannot be determined.

Quantity A is greater.

Quantity B is greater.

Correct answer:

Quantity A is greater.

Explanation:

Since there is an x^2 term in the equation

10x^2-7x=-1

We must acknowledge the possibility of multiple values that satisfy the equation.

Rewrite it so that it is set equal to zero:

10x^2-7x+1=0

For an equation of the form

ax^2+bx+c=0

The solutions for can be found using the quadratic formula:

$x_{1,2}=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$

$x_{1,2}=\frac{-(-7) \pm \sqrt{(-7)^2-4(10)(1)}}{2(10)}=\frac{7 \pm 3}{20}=\frac{1}{5},\frac{1}{2}$

Both values of are positive, so we can't just say x^2 is greater! Compare the two quantities:

$x=\frac{1}{5},x^2=\frac{1}{25};A>B$

$x=\frac{1}{2},x^2=\frac{1}{4};A>B$

Since both values of are greater than zero and less than one, their squares are less than they are.

Quantity A is greater.

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GRE Math : Algebra

Study concepts, example questions & explanations for GRE Math

All GRE Math Resources

Example Questions

Example Question #91 : Expressions

Example Question #92 : Expressions

Example Question #53 : How To Evaluate Algebraic Expressions

Example Question #94 : Expressions

Example Question #91 : Expressions

Example Question #93 : Expressions

Example Question #92 : Expressions

Example Question #93 : Expressions

Example Question #94 : Expressions

Example Question #95 : Expressions

All GRE Math Resources

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