# GRE Math : Algebraic Fractions

## Example Questions

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### Example Question #1 : How To Find Excluded Values

Which of the following are answers to the equation below?

I. -3

II. -2

III. 2

I only

I, II, and III

III only

II and III

II only

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,

The expression  is a difference of squares that can be factored as

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.

The left side factors as follows

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.

### Example Question #1 : How To Find Excluded Values

Which of the following provides the complete solution set for  ?

No solutions

Explanation:

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

### Example Question #3 : Algebraic Fractions

If  then which values of  cannot exist?

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.

Therefore,  and .

### Example Question #4 : Algebraic Fractions

If  then which cannot be an  value?

Explanation:

You cannot take the square root of a negative number.

Setting up the inequality we get:

Solving for  we get:

Therefore any value less than four will not work, .

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

When plugged into  all of the answers give us a value greater than or equal to 0, except for , which gives us .

### Example Question #1 : How To Find Excluded Values

Find the excluded values of the following algebraic fraction

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

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

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

### Example Question #1 : How To Find Inverse Variation

Find the inverse equation of:

Explanation:

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

### Example Question #241 : Gre Quantitative Reasoning

Find the inverse equation of  .

Explanation:

1. Switch the  and  variables in the above equation.

2. Solve for :

### Example Question #1 : Algebraic Fractions

When ,  .

When .

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

Explanation:

If  varies inversely with .

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

Using :

2. Use your new equation  and solve when :

### Example Question #2 : Algebraic Fractions

 x y

If  varies inversely with , what is the value of ?

Explanation:

An inverse variation is a function in the form:  or , where  is not equal to 0.

Substitute each  in .

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

### Example Question #10 : Algebraic Fractions

and  vary inversely. When . When . What does  equal when ?

Explanation:

Because we know  and  vary inversely, we know that  for some .

When .

When .

Therefore, when , we have  so

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