# ISEE Upper Level Quantitative : How to find the solution to an equation

## Example Questions

### Example Question #151 : Equations

Which is the greater quantity?

(a)

(b)

It is impossible to determine which is greater from the information given

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

(b) is the greater quantity

Explanation:

It can be deduced that both  and  are nonnegative, since both are radicands of square roots.

, so

, so

, and

.

### Example Question #151 : How To Find The Solution To An Equation

Which is the greater quantity?

(a)

(b)

(b) is the greater quantity

It is impossible to determine which is greater from the information given

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

Explanation:

By the Zero Product Principle, one of the factors is equal to 0:

which is impossible for any real value of , or

.

By the Zero Product Principle, one of the factors is equal to 0:

which is impossible for any real value of , or

Since  and , it can be determined that .

### Example Question #152 : Equations

Which is the greater quantity?

(a)

(b)

(b) is the greater quantity

It is impossible to determine which is greater from the information given

(a) and (b) are equal

(a) is the greater quantity

(a) is the greater quantity

Explanation:

Between two fractions with the same numerator, the one with the lesser denominator is the greater, so

and .

### Example Question #151 : How To Find The Solution To An Equation

, and  all stand for positive quantities.

Which is the greater quantity?

(a) is the greater quantity

It is impossible to determine which is greater from the information given

(b) is the greater quantity

(a) and (b) are equal

(b) is the greater quantity

Explanation:

Solve the equations for  and  in terms of :

Therefore, we seek to determine which of  and  is greater, bearing in mind that both of these quantities, as well as , must be positive.

We can make the following observation:

Suppose

Then

But if , then

and

Therefore, it must hold that , and  .

### Example Question #153 : Equations

, and  all stand for positive quantities.

Which is the greater quantity?

(a)

(b)

(a) and (b) are equal

(b) is the greater quantity

It is impossible to determine which is greater from the information given

(a) is the greater quantity

(a) is the greater quantity

Explanation:

Solve the equations for  and  in terms of :

and  is positive, so by the properties of inequality,

Solve for :

Explanation:

### Example Question #155 : Equations

Which is the greater quantity?

(a)

(b)

(a) and (b) are equal

(b) is the greater quantity

It is impossible to determine which is greater from the information given

(a) is the greater quantity

(a) and (b) are equal

Explanation:

### Example Question #151 : How To Find The Solution To An Equation

Figure NOT drawn to scale

Above is a straight line on a graph. Which is the greater quantity?

(a)

(b) 18

(a) is the greater quantity

(a) and (b) are equal

(b) is the greater quantity

It is impossible to determine which is greater from the information given

(a) is the greater quantity

Explanation:

If we go from the point (48, 60) to (24, 42), we see that if the first coordinate decreases by 24, the second decreases by 18. Going from  (24, 42) to the point on the -axis, the first coordinate again decreases by 24, so the second coordinate again decreases by 18:

.

### Example Question #157 : Equations

The reciprocal of  is between 2 and 4. Which is the greater quantity?

(a)

(b)

(a) and (b) are equal

(b) is the greater quantity

It is impossible to determine which is greater from the information given

(a) is the greater quantity

It is impossible to determine which is greater from the information given

Explanation:

, so

Also,

, so

Therefore, it possible for

,

,

or

,

making it inconclusive whether  or  is the greater.

Solve for :