How to find the solution to an equation - ISEE Upper Level Quantitative Reasoning
Card 0 of 1416
Define
Which is the greater quantity?
(a) 
(b) 
Define
Which is the greater quantity?
(a)
(b)
(a) To evaluate 
, use the definition for nonnegative values of 
:
(b) To evaluate 
, use the definition for negative values of 
:
(a) To evaluate
(b) To evaluate
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Define
Which is the greater quantity?
(a)
(b)
Define
Which is the greater quantity?
(a)
(b)
(a) To evaluate , use the definition for nonnegative values of :
(b) To evaluate , use the definition for negative values of :
(a) To evaluate
(b) To evaluate
Compare your answer with the correct one above
First, rewrite the quadratic equation in standard form by distributing the 
through the product on the left and collecting all of the terms on the left side:
Use the 
method to factor the quadratic expression 
; we are looking to split the linear term by finding two integers whose sum is 
and whose product is 
. These integers are 
, so:
Set each expression equal to 0 and solve:
or
The solution set is 
.
First, rewrite the quadratic equation in standard form by distributing the
Use the
Set each expression equal to 0 and solve:
or
The solution set is
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First, rewrite the quadratic equation in standard form by FOILing out the product on the left, then collecting all of the terms on the left side:
Use the 
method to split the middle term into two terms; we want the coefficients to have a sum of 1 and a product of 
. These numbers are 
, so we do the following:
Set each expression equal to 0 and solve:
or
The solution set is 
.
First, rewrite the quadratic equation in standard form by FOILing out the product on the left, then collecting all of the terms on the left side:
Use the
Set each expression equal to 0 and solve:
or
The solution set is
Compare your answer with the correct one above
Define
Which is the greater quantity?
(a)
(b)
Define
Which is the greater quantity?
(a)
(b)
(a) To evaluate , use the definition for nonnegative values of :
(b) To evaluate , use the definition for negative values of :
(a) To evaluate
(b) To evaluate
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List all real solutions of the equation
List all real solutions of the equation
By the Zero Product Principle:
, in which case 
,
or
, in which case 
.
The correct choice is 
.
By the Zero Product Principle:
or
The correct choice is
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First, rewrite the quadratic equation in standard form by distributing the through the product on the left and collecting all of the terms on the left side:
Use the method to factor the quadratic expression ; we are looking to split the linear term by finding two integers whose sum is and whose product is . These integers are , so:
Set each expression equal to 0 and solve:
or
The solution set is .
First, rewrite the quadratic equation in standard form by distributing the
Use the
Set each expression equal to 0 and solve:
or
The solution set is
Compare your answer with the correct one above
First, rewrite the quadratic equation in standard form by FOILing out the product on the left, then collecting all of the terms on the left side:
Use the method to split the middle term into two terms; we want the coefficients to have a sum of 1 and a product of . These numbers are , so we do the following:
Set each expression equal to 0 and solve:
or
The solution set is .
First, rewrite the quadratic equation in standard form by FOILing out the product on the left, then collecting all of the terms on the left side:
Use the
Set each expression equal to 0 and solve:
or
The solution set is
Compare your answer with the correct one above
Define
and
.
Evaluate:
Define
and
Evaluate:
First, evaluate 
by using the definition of 
for nonnegative values of 
.
Therefore, 
, so
First, evaluate
Therefore,
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Define
Which is the greater quantity?
(a)
(b)
Define
Which is the greater quantity?
(a)
(b)
(a) To evaluate , use the definition for nonnegative values of :
(b) To evaluate , use the definition for negative values of :
(a) To evaluate
(b) To evaluate
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Define 
as follows:
Which is the greater quantity?
(a) 
(b) 
Define
Which is the greater quantity?
(a)
(b)
(a) 
can be evaluated by using the definition of 
for positive 
:
can be evaluated by using the definition of 
for nonpositive 
:
Add: 
(b) 
can be evaluated by using the definition of 
for nonpositive 
:
(a)
Add:
(b)
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Define as follows:
Which is the greater quantity?
(a)
(b)
Define
Which is the greater quantity?
(a)
(b)
(a) can be evaluated by using the definition of for positive :
can be evaluated by using the definition of for nonpositive :
Add:
(b) can be evaluated by using the definition of for nonpositive :
(a)
Add:
(b)
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Define
and
.
Evaluate:
Define
and
Evaluate:
First, evaluate 
:
Therefore,
which can be evaluated using the definition of 
for nonnegative values of 
:
First, evaluate
Therefore,
which can be evaluated using the definition of
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Which of the following equations has as its solution set 
?
Which of the following equations has as its solution set
The absolute value of a nonnegative number is the number itself; the absolute value of a negative number is its positive opposite.
By substitution, 20 can be seen to be a solution of each of the equations in the four choices.
- true.
20 can be confirmed as a solution to the other three equations similarly. Therefore, the question is essentially to choose the equation with 
as a solution. Substituting 
for 
in each equation:
- true. This is the correct choice.
As for the other three:
- false.
The other two equations can be similarly proved to not have 
as a solution.
The absolute value of a nonnegative number is the number itself; the absolute value of a negative number is its positive opposite.
By substitution, 20 can be seen to be a solution of each of the equations in the four choices.
20 can be confirmed as a solution to the other three equations similarly. Therefore, the question is essentially to choose the equation with
As for the other three:
The other two equations can be similarly proved to not have
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Give the solution set of the equation:
Give the solution set of the equation:
Either
or 
,
so we solve the equations separately:
or
The solution set is 
Either
so we solve the equations separately:
or
The solution set is
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List all real solutions of the equation
List all real solutions of the equation
By the Zero Product Principle:
, in which case ,
or
, in which case .
The correct choice is .
By the Zero Product Principle:
or
The correct choice is
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Give the solution set of the equation
Give the solution set of the equation
Since it is impossible for the absolute value of a number to be negative, the equation has no solution.
Since it is impossible for the absolute value of a number to be negative, the equation has no solution.
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Give the solution set of the equation
Give the solution set of the equation
Since it is impossible for the absolute value of a number to be negative, the equation has no solution.
Since it is impossible for the absolute value of a number to be negative, the equation has no solution.
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Give the solution set of the equation
Give the solution set of the equation
Since it is impossible for the absolute value of a number to be negative, the equation has no solution.
Since it is impossible for the absolute value of a number to be negative, the equation has no solution.
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Give the solution set of the equation
Give the solution set of the equation
Since it is impossible for the absolute value of a number to be negative, the equation has no solution.
Since it is impossible for the absolute value of a number to be negative, the equation has no solution.
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