Card 0 of 136
Define an operation on the set of real numbers as follows:
For any two real numbers
Evaluate the expression
Substitute in the expression:
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Solve for .
To solve for x we need to make two separate equations. Since it has absolute value bars around it we know that the inside can equal either 7 or -7 before the asolute value is applied.
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Simplify the following expression:
To simplify, we must first simplify the absolute values.
Now, combine like terms:
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The absolute value of a negative can be positive or negative. True or false?
The absolute value of a number is the points away from zero on a number line.
Since this is a countable value, you cannot count a negative number.
This makes all absolute values positive and also make the statement above false.
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Consider the quadratic equation
Which of the following absolute value equations has the same solution set?
Rewrite the quadratic equation in standard form by subtracting from both sides:
Factor this as
where the squares represent two integers with sum and product 14. Through some trial and error, we find that
and
work:
By the Zero Product Principle, one of these factors must be equal to 0.
If then
;
if then
.
The given equation has solution set , so we are looking for an absolute value equation with this set as well.
This equation can take the form
This can be rewritten as the compound equation
Adding to both sides of each equation, the solution set is
and
Setting these numbers equal in value to the desired solutions, we get the linear system
Adding and solving for :
Backsolving to find :
The desired absolute value equation is .
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What is the value of: ?
Step 1: Evaluate ...
Step 2: Apply the minus sign inside the absolute value to the answer in Step 1...
Step 3: Define absolute value...
The absolute value of any value is always positive, unless there is an extra negation outside (sometimes)..
Step 4: Evaluate...
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Solve:
Divide both sides by negative three.
Since the lone absolute value is not equal to a negative, we can continue with the problem. Split the equation into its positive and negative components.
Evaluate the first equation by subtracting one on both sides, and then dividing by two on both sides.
Evaluate the second equation by dividing a negative one on both sides.
Subtract one on both sides.
Divide by 2 on both sides.
The answers are:
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Define an operation as follows:
For all real numbers ,
If , which is a possible value of
?
, so
can be rewritten as
Therefore, either or
. The correct choice is
.
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Define .
How many values are in the solution set of the equation ?
We can rewrite this function as a piecewise-defined function by examining three different intervals of -values.
If , then
and
,
and this part of the function can be written as
If , then
and
,
and this part of the function can be written as
If , then
and
,
and this part of the function can be written as
The function can be rewritten as
As can be seen from the rewritten definition, every value of in the interval
is a solution of
, so the correct response is infinitely many solutions.
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Define .
How many values are in the solution set of the equation ?
We can rewrite this function as a piecewise-defined function by examining three different intervals of -values.
If , then
and
,
and this part of the function can be written as
If under this definition, then
However, , so this is a contradiction.
If , then
and
,
and this part of the function can be written as
This yields no solutions.
If , then
and
,
and this part of the function can be written as
If under this definition, then
However, , so this is a contradiction.
has no solution.
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Define .
Evaluate .
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Define .
Order from least to greatest:
, or, equivalently,
From least to greatest, the values are
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Define an operation as follows:
For all real numbers ,
Evaluate .
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Consider the quadratic equation
Which of the following absolute value equations has the same solution set?
Rewrite the quadratic equation in standard form by subtracting from both sides:
Solve this equation using the method. We are looking for two integers whose sum is
and whose product is
; by trial and error we find they are
,
. The equation becomes
Solving using grouping:
By the Zero Product Principle, one of these factors must be equal to 0.
Either
Or
The given quadratic equation has solution set , so we are looking for an absolute value equation with this set as well.
This equation can take the form
This can be rewritten as the compound equation
Adding to both sides of each equation, the solution set is
and
Setting these numbers equal in value to the desired solutions, we get the linear system
Adding and solving for :
Backsolving to find :
The desired absolute value equation is .
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Give the solution set of the inequality:
To solve an absolute value inequality, first isolate the absolute value expression, which can be done here by subtracting 35 from both sides:
There is no need to go further. The absolute value of any number is always greater than or equal to 0, so, regardless of the value of ,
.
Therefore, the solution set is the set of all real numbers.
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Solve .
First we need to isolate the absolute value term. We do with using some simple algebra:
Now we solve two equations, one with the right side of the equation positive, one negative. Let's start with the positive:
And now the negative:
So our answers are:
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Solve .
First, we have to isolate the absolute value:
Let's take a look at our equation right now. It's saying that the absolute value has to be a negative number, which isn't possible. So no solutions exist.
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Solve .
First, we need to isolate the absolute value:
Because the equation is set equal to , we can drop the absolute value symbols and solve normally:
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Solve:
Add 3 on both sides.
Divide by 25 on both sides.
Recall that an absolute value cannot have a negative value. There is no x-value that will equal to the right term.
The answer is:
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Solve: .
Because the absolute value term is "less than" the other side of the equation, we can rewrite the problem like this:
This eliminates the absolute value. Remember, when an operation is performed, it must be performed on all three sets of terms. When we add to each side, we end up with:
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