GRE Math : Algebraic Fractions

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Example Questions

Example Question #1 : How To Find Excluded Values

Which of the following are answers to the equation below?

$\frac{x^2-4}{x^2+5x+6}=0$

I. -3

II. -2

III. 2

Possible Answers:

II only

II and III

I, II, and III

I only

III only

Correct answer:

III only

Explanation:

Given a fractional algebraic equation with variables in the numerator and denominator of one side and the other side equal to zero, we rely on a simple concept. Zero divided by anything equals zero. That means we can focus in on what values make the numerator (the top part of the fraction) zero, or in other words,

x^2-4=0

The expression x^2-4 is a difference of squares that can be factored as

(x-2)(x+2)=0

Solving this for gives either or . That means either of these values will make our numerator equal zero. We might be tempted to conclude that both are valid answers. However, our statement earlier that zero divided by anything is zero has one caveat. We can never divide by zero itself. That means that any values that make our denominator zero must be rejected. Therefore we must also look at the denominator.

x^2+5x+6=0

The left side factors as follows

(x+3)(x+2)=0

This means that if is or , we end up dividing by zero. That means that cannot be a valid solution, leaving as the only valid answer. Therefore only #3 is correct.

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Example Question #1 : How To Find Excluded Values

Which of the following provides the complete solution set for |2-z|>0 ?

Possible Answers:

z>2

z=2

z<2

$\dpi{100} z|z\neq 2$

No solutions

Correct answer:

$\dpi{100} z|z\neq 2$

Explanation:

The absolute value will always be positive or 0, therefore all values of z will create a true statement as long as $2-z\neq 0$ . Thus all values except for 2 will work.

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Example Question #1 : Algebraic Fractions

If $f(x)=\frac{(x^2-25)}{(3x^2+4x-7)}$ then which values of cannot exist?

Possible Answers:

x=1

x=-1

x=-7

x=-5

x=5

Correct answer:

x=1

Explanation:

The denominator of a fraction can never be equal to 0.

Therefore, to find out what x cannot be equal to, we must factor the denominator, and determine what values of x would make it equal to 0.

3x^2+4x-7=(3x+7)(x-1)

Therefore, $x\neq-\frac{7}{3}$ and $x\neq1$ .

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Example Question #2 : How To Find Excluded Values

If $f(x)=\sqrt{x^2-16}$ then which cannot be an value?

Possible Answers:

x=-5.2

x=3.5

x=-6.7

x=16

x=4

Correct answer:

x=3.5

Explanation:

You cannot take the square root of a negative number.

Setting up the inequality we get:

$x^2-16\ge 0$

Solving for we get:

$x^2\ge 16$

$x\ge \sqrt{16}$

$x\ge 4$

Therefore any value less than four will not work, x=3.5 .

Another approach is to plug in each of the possible values.

When plugged into $f(x)=\sqrt{x^2-16}$ all of the answers give us a value greater than or equal to 0, except for x=3.5 , which gives us $\sqrt{-3.75}$ .

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Example Question #2 : How To Find Excluded Values

Find the excluded values of the following algebraic fraction

$\frac{2x^2-5x-12}{x^2-11x+28}$

Possible Answers:

$x\neq7$

$x\neq4,7$

The numerator cancels all the binomials in the denomniator so ther are no excluded values.

$x\neq4$

$x\neq-11$

Correct answer:

$x\neq4,7$

Explanation:

To find the excluded values of a algebraic fraction you need to find when the denominator is zero. To find when the denominator is zero you need to factor it. This denominator factors into

(x-7)(x-4)

so this is zero when x=4,7 so our answer is

$x\neq4,7$

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Example Question #1 : Algebraic Fractions

Find the inverse equation of:

3y-2x=20

Possible Answers:

$\frac{1}{2}x+10=y$

$\frac{2x+20}{3}=y$

$\frac{2x-20}{3}=y$

$\frac{3x-20}{2}=y$

Correct answer:

$\frac{3x-20}{2}=y$

Explanation:

To solve for an inverse, we switch x and y and solve for y. Doing so yields:

$\frac{3x-20}{2}=y$

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Example Question #2 : How To Find Inverse Variation

Find the inverse equation of 5x-4y=18 .

Possible Answers:

x=y

$x=\frac{4y+18}{5}$

$y=\frac{4x+5}{18}$

$y=\frac{x+18}{4}$

$y=\frac{4x+18}{5}$

Correct answer:

$y=\frac{4x+18}{5}$

Explanation:

5x-4y=18

1. Switch the and variables in the above equation.

5y-4x=18

2. Solve for :

5y-4x=18

5y-4x+4x=18+4x

5y=4x+18

$\frac{5y}{5}=\frac{4x+18}{5}$

$y=\frac{4x+18}{5}$

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Example Question #3 : How To Find Inverse Variation

$\begin{matrix} & \x\ & \y\ & \\ & \\ & \end{matrix}$ When x=2 , y=18 .

When x=9 , y=4 .

If varies inversely with , what is the value of when x=12 ?

Possible Answers:

y=1.5

y=3

y=x

y=6

y=1

Correct answer:

y=3

Explanation:

If varies inversely with , $y=\frac{K}{x}$ .

1. Using any of the two x,y combinations given, solve for :

Using (2,18) :

$18=\frac{K}{2}$

K=36

2. Use your new equation $y=\frac{36}{x}$ and solve when x=12 :

$y=\frac{36}{12}=3$

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Example Question #1 : How To Find Inverse Variation

x	y

If varies inversely with , what is the value of ?

Possible Answers:

Correct answer:

Explanation:

An inverse variation is a function in the form: xy = k or $y = \frac{k}{x}$ , where is not equal to 0.

Substitute each $\left ( x,y \right )$ in xy = k .

5(4.8) = 24

6.4(3.75) = 24

20(1.2) = 24

Therefore, the constant of variation, , must equal 24. If varies inversely as , must equal 24. Solve for .

3n = 24

n = 8

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Example Question #3 : How To Find Inverse Variation

and vary inversely. When x=1 , y=14 . When x=2 , y=7 . What does equal when x=4 ?

Possible Answers:

y=3.5

y=1

y=4.67

y=21

y=4

Correct answer:

y=3.5

Explanation:

Because we know and vary inversely, we know that $x=\frac{k}{y}$ for some .

When x=1 , y=14 . $1=\frac{14}{14}$ .

When x=2 , y=7 . $2=\frac{14}{7}$ .

Therefore, when x=4 , we have $4=\frac{14}{y}$ so $y=\frac{14}{4}=3.5$

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GRE Math : Algebraic Fractions

Study concepts, example questions & explanations for GRE Math

All GRE Math Resources

Example Questions

Example Question #1 : How To Find Excluded Values

Example Question #1 : How To Find Excluded Values

Example Question #1 : Algebraic Fractions

Example Question #2 : How To Find Excluded Values

Example Question #2 : How To Find Excluded Values

Example Question #1 : Algebraic Fractions

Example Question #2 : How To Find Inverse Variation

Example Question #3 : How To Find Inverse Variation

Example Question #1 : How To Find Inverse Variation

Example Question #3 : How To Find Inverse Variation

All GRE Math Resources

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