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

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

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Example Questions

Example Question #21 : Equations

 refers to the least integer greater than or equal to .

 and  are integers. 

Which is the greater quantity?

(a) 

(b) 

Possible Answers:

(a) is greater.

(a) and (b) are equal.

(b) is greater.

It is impossible to tell from the information given.

Correct answer:

(a) is greater.

Explanation:

(a) Since  is an integer, .

Since  is an integer, .

(b) By closure,  is an integer, so 

.

(a) is the greater quantity.

Example Question #21 : Algebraic Concepts

 refers to the greatest integer less than or equal to .

 and  are integers.

Which is greater?

(a) 

(b) 

Possible Answers:

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

Correct answer:

(b) is greater.

Explanation:

(a) Since  is an integer, .

Since  is an integer, .

(b) By closure,  is an integer, so 

.

This makes (b) greater.

Example Question #23 : Equations

 

Which is the greater quantity?

(a) 

(b)

Possible Answers:

(a) is greater.

(a) and (b) are equal.

(b) is greater.

It cannot be determined from the information given.

Correct answer:

(a) and (b) are equal.

Explanation:

Substitute  and, subsequently, :

Factor as , replacing the two question marks with integers whose product is  and whose sum is . These integers are .

Break this up into two equations, replacing  for :

or

This has no solution, since  must be nonnegative.

is the only solution, so (a) and (b) must be equal.

 

Example Question #22 : Algebraic Concepts

Consider the line through points  and .

Which is the greater quantity?

(a) The -coordinate of the -intercept of this line

(b) The -coordinate of the -intercept of this line

Possible Answers:

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

Correct answer:

(a) is greater.

Explanation:

The slope of this line is 

.

We will use the point-slope form of the line, with this slope and point :

The -coordinate of the -intercept of this line can be found by substituting  and solving for :

 

The -coordinate of the -intercept of this line can be found by substituting  and solving for :

This makes (a) the greater quantity.

Example Question #25 : Equations

The slope of a line is 2; the line does not pass through the origin.

Which is the greater quantity?

(a) The -coordinate of the -intercept 

(b) The -coordinate of the -intercept 

Possible Answers:

It is impossible to tell from the information given.

(a) and (b) are equal.

(a) is greater.

(b) is greater.

Correct answer:

It is impossible to tell from the information given.

Explanation:

Let  be the - and -intercepts, respectively. We know that the line does not pass through the origin - so .

Then the slope is:

Either  or  can be the greater. For example, if , then , and if , then 

Example Question #26 : Equations

.

Which is the greater quantity?

(a) 

(b)

Possible Answers:

It is impossible to tell from the information given.

(b) is greater.

(a) and (b) are equal.

(a) is greater.

Correct answer:

(a) and (b) are equal.

Explanation:

, so substitute and use the power of a power rule.

This makes (a) and (b) equal.

Example Question #27 : Equations

Which is the greater quantity?

(a) The slope of the line of the equation 

(b) The slope of the line of the equation 

Possible Answers:

(a) is greater.

(a) and (b) are equal.

(b) is greater.

It is impossible to tell from the information given.

Correct answer:

(a) is greater.

Explanation:

Both equations are in slope-intercept form, so compare the coefficients of . The coefficients in (a) and (b) are 5 and 4, respectively, so these are the slopes of the lines. The line in (a) has the greater slope.

Example Question #23 : Algebraic Concepts

Which is the greater quantity?

(a) 

(b) 

Possible Answers:

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

(a) and (b) are equal.

Correct answer:

It is impossible to tell from the information given.

Explanation:

Using two different cases, we show that it is impossible to tell which is greater.

Case 1: . Then , and .

Case 2: . Then , and .

Example Question #29 : Equations

Which is the greater quantity?

(a) 

(b) 

Possible Answers:

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

Correct answer:

(a) is greater.

Explanation:

To solve the system of equations, add the left and right sides of the equation separately:

  

Divide:

 

Substitute to get :

 

 is greater.

Example Question #24 : Algebraic Concepts

Which is the greater quantity?

(a) 

(b) 

Possible Answers:

(a) is greater

(b) is greater

It is impossible to tell from the information given

(a) and (b) are equal

Correct answer:

It is impossible to tell from the information given

Explanation:

We show that it is possible for either  or  to be the greater by giving one of each case.

Case 1: . Then , so 

Case 2: . Then , so 

In Case 1, ; in Case 2, 

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