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Example Question #2197 : Mathematical Relationships And Basic Graphs
Solve for .
and
and
and
and
Solve for given
When given an absolute value equation recognize there are often multiple solutions. The reason why is best exhibited in a simpler example:
Given solve for values
of that make this statement true. When you taken an absolute value of something you always end up with the positive number so both
and
would make this statement true. The solutions can also be written as ±
.
In the case of the more complicated equation , however before proceeding let's simplify this equation a little as there are two
terms that can be added together within the absolute value. These are
and
.
When added together this gives ,
thereby giving you
For the same reason as shown in the case of there are potentially two solutions to this equation, which are shown by
as an absolute value will always end up creating a positive result. Make sure you apply the parentheses to only the portion of the equation with the absolute value otherwise your answer will be incorrect.
To simplify the absolute value we must look at each of these cases:
Let's start with the positive case:
Just like a normal equation with one unknown we will simplify it by isolating by itself. This is first done by combining like terms and getting
on one side of the equation.
We can first subtract from
which is
, this gives you:
Next we can subtract from both sides of the equation.
Dividing both sides by gives you a final answer of
this however can be simplified to
as both the numerator and denominator are divisible by
.
so
To check this solution it must be substituted in the original absolute value for and if it's a correct answer you'll end up with a true statement, so
,
Simplify the equation by multiplying by
and
by
This leaves our equation with
Next add to both sides of the equation:
In order to simplify you must find a common denominator, which is most easily done by multiplying
.
This leaves:
Similarly a common denominator is found for and
by multiplying
by
which gives you
this leaves:
simplifying further gives you
which is a true statement, so
is a valid solution
Next let's solve for the negative case:
Distribute the negative sign, which is just to make calculations easier and you'll get:
Combine like terms:
This can be simplified to as both the numerator and denominator are divisible by
, therefore you final answer is
Checking this solution is done just as you did for the previous solution obtained.
Given
substitute in for
Multiplying by
gives
Multiplying gives
So the equation simplifies to:
Next can be added to both sides of the equation giving:
Now common denominators must be found for and
The common denominator for is found by multiplying
by
. This gives you
. This can then be simplified through addition and gives
.
The common denominator of is found by multiplying
and
so
The simplified equation becomes
Through dividing by
you get
and the absolute value of
is
, so you get
. This is true statement
so is also a valid solution.
Example Question #2191 : Mathematical Relationships And Basic Graphs
Solve for :
To solve absolute value equations, we must understand that the absoute value function makes a value positive. So when we are solving these problems, we must consider two scenarios, one where the value is positive and one where the value is negative.
and
This gives us:
and
However, this question has an outside of the absolute value expression, in this case
. Thus, any negative value of
will make the right side of the equation equal to a negative number, which cannot be true for an absolute value expression. Thus,
is an extraneous solution, as
cannot equal a negative number.
Our final solution is then
Example Question #71 : Solving Absolute Value Equations
Which values of provide the full solution set for the inequality:
Example Question #4861 : Algebra Ii
Refer to the above figure.
Which of the following functions is graphed?
Below is the graph of :
The given graph is the graph of reflected in the
-axis, then translated up 6 units. This graph is
, where
.
The function graphed is therefore
Example Question #2 : Graphing Absolute Value Functions
Refer to the above figure.
Which of the following functions is graphed?
Below is the graph of :
The given graph is the graph of reflected in the
-axis, then translated left 2 units (or, equivalently, right
units. This graph is
, where
.
The function graphed is therefore
Example Question #3 : Graphing Absolute Value Functions
Refer to the above figure.
Which of the following functions is graphed?
The correct answer is not given among the other responses.
Below is the graph of :
The given graph is the graph of translated by moving the graph 7 units left (that is,
unit right) and 2 units down (that is,
units up)
The function graphed is therefore
where
. That is,
Example Question #3 : Graphing Absolute Value Functions
What is the equation of the above function?
The formula of an absolute value function is where m is the slope, a is the horizontal shift and b is the vertical shift. The slope can be found with any two adjacent integer points, e.g.
and
, and plugging them into the slope formula,
, yielding
. The vertical and horizontal shifts are determined by where the crux of the absolute value function is. In this case, at
, and those are your a and b, respectively.
Example Question #4 : Graphing Absolute Value Functions
Give the vertex of the graph of the function .
None of the other choices gives the correct response.
Let
The graph of this basic absolute value function is a "V"-shaped graph with a vertex at the origin, or the point with coordinates . In terms of
,
The graph of this function can be formed by shifting the graph of left 6 units (
) and down 7 units (
). The vertex is therefore located at
.
Example Question #2 : Graphing Absolute Value Functions
Give the vertex of the graph of the function .
None of the other choices gives the correct response.
Let
The graph of this basic absolute value function is a "V"-shaped graph with a vertex at the origin, or the point with coordinates . In terms of
,
,
or, alternatively written,
The graph of is the same as that of
, after it shifts 10 units left (
), it flips vertically (negative symbol), and it shifts up 10 units (the second
). The flip does not affect the position of the vertex, but the shifts do; the vertex of the graph of
is at
.
Example Question #7 : Graphing Absolute Value Functions
Which of the following absolute value functions is represented by the following graph?
The equation cannot be determined from the graph.
The equation can be determined from the graph by following the rules of transformations; the base equation is:
The graph of this base equation is:
When we compare our graph to the base equation graph, we see that it has been shifted right 3 units, up 1 unit, and our graph has been stretched vertically by a factor of 2. Following the rules of transformations, the equation for our graph is written as:
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