How to find the solution to an equation - ISEE Upper Level Quantitative Reasoning
Card 1 of 1416
Give the solution set of the equation
Give the solution set of the equation
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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
Tap to reveal answer
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
Tap to reveal answer
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
Tap to reveal answer
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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Define 
as follows:
Which is the greater quantity?
(a) 
(b) 
Define
Which is the greater quantity?
(a)
(b)
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(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
Which is the greater quantity?
(a) 
(b) 
Define
Which is the greater quantity?
(a)
(b)
Tap to reveal answer
(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
and
.
Evaluate:
Define
and
Evaluate:
Tap to reveal answer
First, evaluate 
by using the definition of 
for nonnegative values of 
.
Therefore, 
, so
First, evaluate
Therefore,
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Define as follows:
Which is the greater quantity?
(a)
(b)
Define
Which is the greater quantity?
(a)
(b)
Tap to reveal answer
(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
Which is the greater quantity?
(a)
(b)
Define
Which is the greater quantity?
(a)
(b)
Tap to reveal answer
(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
and
.
Evaluate:
Define
and
Evaluate:
Tap to reveal answer
First, evaluate by using the definition of for nonnegative values of .
Therefore,
, so
First, evaluate
Therefore,
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Define
and
.
Evaluate:
Define
and
Evaluate:
Tap to reveal answer
First, evaluate by using the definition of for nonnegative values of .
Therefore,
, so
First, evaluate
Therefore,
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Define as follows:
Which is the greater quantity?
(a)
(b)
Define
Which is the greater quantity?
(a)
(b)
Tap to reveal answer
(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
Which is the greater quantity?
(a)
(b)
Define
Which is the greater quantity?
(a)
(b)
Tap to reveal answer
(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
and
.
Evaluate:
Define
and
Evaluate:
Tap to reveal answer
First, evaluate by using the definition of for nonnegative values of .
Therefore,
, so
First, evaluate
Therefore,
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List all real solutions of the equation
List all real solutions of the equation
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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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Solve for 
, giving all real solutions:
Solve for
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Write the equation in standard form:
can be factored out of each term:
Factor the trinomial by writing 
, replacing the question marks with two integers with product 
and sum 
. These integers are 
, so the above becomes
.
We can disregard the 
, as it does not contribute a solution. Set each of the other two factors equal to 
and solve separately:
The solution set is 
.
Write the equation in standard form:
Factor the trinomial by writing
We can disregard the
The solution set is
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Define
and
.
Evaluate:
Define
and
Evaluate:
Tap to reveal answer
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
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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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Define
and
.
Evaluate:
Define
and
Evaluate:
Tap to reveal answer
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
Tap to reveal answer
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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