How to find out when an equation has no solution - GRE Quantitative Reasoning

First, distribute, making sure to watch for negatives.

Combine like terms.

Subtract 7x from both sides.

Add 18 on both sides and be careful adding integers.

First, distribute the to the terms inside the parentheses.

Add 6x to both sides.

This is false for any value of . Thus, there is no solution.

By definition, the absolute value of an expression can never be less than 0. Therefore, there are no solutions to the above expression.

Expand out into .

Since , it can be seen that

so Quantity B is greater.

Percent growth is given as:

For Beetleton, this can be expressed as (in terms of billions of US dollars):

Let us write down what we are told in mathematical terms, designating the smaller integer as and the larger integer as .

The sum of the two integers is :

And the larger integer is % greater than the smaller integer:

Writing the first equation in terms of gives:

Which allows us to find :

Thus, the positive difference between the two is found as

To solve this problem, expand each function described by Quantities A and B:

Quantity A:

Quantity B:

Now note that Quantities A and B only differ in that Quantity A is greater by .

Since we are told that is greater than and thus always positive, Quantity A must be greater than Quantity B for all possible values of .

Rather than manually finding common denominators and adding the fractions together, realize that

Since

Quantity A must be greater, and this can be seen without actually calculating its value.

Column B is greater for positive numbers.

The columns are equal for 0.

Column A is greater for negative numbers.

Because our answer changes depending on the value inserted, we cannot determine the relationship.

Substituting 3 in for x, you will get 0 in the denominator of the fraction. It is not possible to have 0 be the denominator for a fraction so there is no possible solution to this equation.

We are given that y = 32. Plug this value of y into the second equation.

32 = x2 – 4

36 = x2

x = +/– 6.

Next find a value for Quantity A:

y/7 = 32/7

This number is less than +6, but more than –6. Thus, the relationship cannot be determined from the information given.

Rather than manually finding common denominators and adding the fractions together, realize that

Since

Quantity A must be greater, and this can be seen without actually calculating its value.

Column B is greater for positive numbers.

The columns are equal for 0.

Column A is greater for negative numbers.

Because our answer changes depending on the value inserted, we cannot determine the relationship.

Substituting 3 in for x, you will get 0 in the denominator of the fraction. It is not possible to have 0 be the denominator for a fraction so there is no possible solution to this equation.

We are given that y = 32. Plug this value of y into the second equation.

32 = x2 – 4

36 = x2

x = +/– 6.

Next find a value for Quantity A:

y/7 = 32/7

This number is less than +6, but more than –6. Thus, the relationship cannot be determined from the information given.