# GRE Math : How to find the solution to an inequality with division

## Example Questions

### Example Question #1 : How To Find The Solution To An Inequality With Division

For how many positive integers, , is it true that

More than

None

Explanation:

Since  is positive, we can divide both sides of the inequality by :

or .

### Example Question #5 : How To Find The Solution For A System Of Equations

Solve for .

Explanation:

For the second equation, solve for  in terms of .

Plug this value of y into the first equation.

### Example Question #2 : How To Find The Solution To An Inequality With Division

Solve for the -intercept:

Explanation:

Don't forget to switch the inequality direction if you multiply or divide by a negative.

Now that we have the equation in slope-intercept form, we can see that the y-intercept is 6.

### Example Question #3 : How To Find The Solution To An Inequality With Division

Solve for :

Explanation:

Begin by adding  to both sides, this will get the variable isolated:

Or...

Next, divide both sides by :

Notice that when you divide by a negative number, you need to flip the inequality sign!

### Example Question #4 : How To Find The Solution To An Inequality With Division

Quantity A:

The smallest possible value for

Quantity B:

The smallest possible value for

Which of the following is true?

Quantity A is larger.

A comparison cannot be detemined from the given information.

Quantity B is larger.

The two quantities are equal.

Quantity A is larger.

Explanation:

Recall that when you have an absolute value and an inequality like

,

this is the same as saying that  must be between  and .  You can rewrite it:

To solve this, you just apply your modifications to each and every part of the inequality.

First, subtract :

Then, divide by :

Next, do the same for the other equation.

becomes...

Then, subtract :

Then, divide by :

Thus, the greatest value for  is , while the smallest value for  is .  Therefore, quantity A is greater.

### Example Question #5 : How To Find The Solution To An Inequality With Division

Each of the following is equivalent to

xy/z * (5(x + y))  EXCEPT:

5x²y + 5xy²/z

xy(5x + 5y)/z

xy(5y + 5x)/z

5x² + y²/z

5x² + y²/z

Explanation:

Choice a is equivalent because we can say that technically we are multiplying two fractions together: (xy)/z and (5(x + y))/1.  We multiply the numerators together and the denominators together and end up with xy (5x + 5y)/z.  xy (5y + 5x)/z is also equivalent because it is only simplifying what is inside the parentheses and switching the order- the commutative property tells us this is still the same expression.  5x²y + 5xy²/z is equivalent as it is just a simplified version when the numerators are multiplied out.  Choice 5x² + y²/z is not equivalent because it does not account for all the variables that were in the given expression and it does not use FOIL correctly.

### Example Question #6 : How To Find The Solution To An Inequality With Division

Let S be the set of numbers that contains all of values of x such that 2x + 4 < 8. Let T contain all of the values of x such that -2x +3 < 8. What is the sum of all of the integer values that belong to the intersection of S and T?

-3

-7

2

-2

0

-2

Explanation:

First, we need to find all of the values that are in the set S, and then we need to find the values in T. Once we do this, we must find the numbers in the intersection of S and T, which means we must find the values contained in BOTH sets S and T.

S contains all of the values of x such that 2x + 4 < 8. We need to solve this inequality.

2x + 4 < 8

Subtract 4 from both sides.

2x < 4

Divide by 2.

x < 2

Thus, S contains all of the values of x that are less than (but not equal to) 2.

Now, we need to do the same thing to find the values contained in T.

-2x + 3 < 8

Subtract 3 from both sides.

-2x < 5

Divide both sides by -2. Remember, when multiplying or dividing an inequality by a negative number, we must switch the sign.

x > -5/2

Therefore, T contains all of the values of x that are greater than -5/2, or -2.5.

Next, we must find the values that are contained in both S and T. In order to be in both sets, these numbers must be less than 2, but also greater than -2.5. Thus, the intersection of S and T consists of all numbers between -2.5 and 2.

The question asks us to find the sum of the integers in the intersection of S and T. This means we must find all of the integers between -2.5 and 2.

The integers between -2.5 and 2 are the following: -2, -1, 0, and 1. We cannot include 2, because the values in S are LESS than but not equal to 2.

Lastly, we add up the values -2, -1, 0, and 1. The sum of these is -2.

### Example Question #41 : Inequalities

What is the solution set of the inequality  ?

Explanation:

We simplify this inequality similarly to how we would simplify an equation

Thus

### Example Question #42 : Inequalities

What is a solution set of the inequality ?