Absolute Value - Algebra 2
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Solve:
Solve:
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To solve:
1. To clear the absolute value sign, we must split the equation to two possible solution. One solution for the possibility of a positive sign and another for the possibility of a negative sign:
or 
2. solve each equation for 
:
or 
To solve:
1. To clear the absolute value sign, we must split the equation to two possible solution. One solution for the possibility of a positive sign and another for the possibility of a negative sign:
2. solve each equation for
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Solve the following equation for b:
Solve the following equation for b:
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Solve the following equation for b:
Let's begin by subtracting 6 from both sides:
Next, we can get rid of the absolute value signs and make our two equations:
Next, add 13 to both sides
And divide by 5:
or
Next, add 13 to both sides
And divide by 5:
So our answers are
Solve the following equation for b:
Let's begin by subtracting 6 from both sides:
Next, we can get rid of the absolute value signs and make our two equations:
Next, add 13 to both sides
And divide by 5:
or
Next, add 13 to both sides
And divide by 5:
So our answers are
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Solve for x:
Solve for x:
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Since the absolute value function only produces positive answers, an absolute value can never be equal to a negative number.
Since the absolute value function only produces positive answers, an absolute value can never be equal to a negative number.
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Solve the following equation:
Solve the following equation:
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To solve this you need to set up two different equations then solve for x.
The first one is:
where 
The other equations is:
where 
To solve this you need to set up two different equations then solve for x.
The first one is:
The other equations is:
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Find values of 
which satisfy,
Find values of
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Recall the general definition of absolute value,
We will attempt two cases for the absolute value,
Case 1:
Start by isolating the abolute value term,
Replace 
with 
and solve for 
:
Case 2:
Replace 
with 
and solve for 
:
Testing the Solutions
Whenever solving equations with an absolute value it is crucuial to check if the solutions work in the original equation. It often occurs that one or both of the solutions will not satisfy the original equation.
For instance, if we test 
in the original equation we get,
This is clearly not true, since both sides are not equal, this rules out 
as a solution. Similiarly, using 
you can show that it also fails. Therefore, there is no solution.
Recall the general definition of absolute value,
We will attempt two cases for the absolute value,
Case 1:
Start by isolating the abolute value term,
Replace
Case 2:
Replace
Testing the Solutions
Whenever solving equations with an absolute value it is crucuial to check if the solutions work in the original equation. It often occurs that one or both of the solutions will not satisfy the original equation.
For instance, if we test
This is clearly not true, since both sides are not equal, this rules out
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What are all the possible values of 
that fulfill the equation below?
What are all the possible values of
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If 
, then 
or 
Solve each of those equations to find the possible values for x.
or 
If
Solve each of those equations to find the possible values for x.
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Solve for 
.
Solve for
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When dealing with absolute value equations, we need to deal with negative values as well.
Subtract 
on both sides.
Distribute the negative sign to each term in the parenthesEs.
Add 
on both sides.
Divide both sides by 
.
Answers are 
When dealing with absolute value equations, we need to deal with negative values as well.
Answers are
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Solve for 
.
Solve for
Tap to reveal answer
When dealing with absolute value equations, we need to deal with negative values as well.
Add 
on both sides.
Subtract 
on both sides.
Distribute the negative sign to each term in the parentheses.
Add 
on both sides.
Divide 
on both sides.
Answers are 
When dealing with absolute value equations, we need to deal with negative values as well.
Answers are
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Solve for 
.
Solve for
Tap to reveal answer
When dealing with absolute value equations, we need to deal with negative values as well.
Subtract 
on both sides.
Divide 
on both sides.
When dealing with absolute value equations, we need to deal with negative values as well.
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Tap to reveal answer
To solve an absolute value equation, one must first rewrite the problem two different ways.
One is
and the other is
.
Then, solve each equation for v.
Your answers are 8 and -5.
To solve an absolute value equation, one must first rewrite the problem two different ways.
One is
and the other is
Then, solve each equation for v.
Your answers are 8 and -5.
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Solve the following equation:
Solve the following equation:
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To solve absolute value equations, first we must isolate the absolute value:
Now that the absolute value is the only thing on the left hand side of the equation, we can solve for x.
Keep in mind that the absolute value makes whatever is inside of it a positive value. This means that 
and 
are both valid in solving the equation.
Setting both of these equal to 3 and solving for x, we get
As a check, plug each of these solutions back into the original equation and see if both solutions are valid!
To solve absolute value equations, first we must isolate the absolute value:
Now that the absolute value is the only thing on the left hand side of the equation, we can solve for x.
Keep in mind that the absolute value makes whatever is inside of it a positive value. This means that
Setting both of these equal to 3 and solving for x, we get
As a check, plug each of these solutions back into the original equation and see if both solutions are valid!
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What is the equation of the above function?
What is the equation of the above function?
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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.
The formula of an absolute value function is
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Refer to the above figure.
Which of the following functions is graphed?
Refer to the above figure.
Which of the following functions is graphed?
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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,
Below is the graph of
The given graph is the graph of
The function graphed is therefore
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Refer to the above figure.
Which of the following functions is graphed?
Refer to the above figure.
Which of the following functions is graphed?
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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
Below is the graph of
The given graph is the graph of
The function graphed is therefore
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Refer to the above figure.
Which of the following functions is graphed?
Refer to the above figure.
Which of the following functions is graphed?
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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
Below is the graph of
The given graph is the graph of
The function graphed is therefore
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Which of the following absolute value functions is represented by the following graph?
Which of the following absolute value functions is represented by the following graph?
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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:
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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Give the vertex of the graph of the function 
.
Give the vertex of the graph of the function
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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 
.
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
The graph of this function can be formed by shifting the graph of
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Give the vertex of the graph of the function 
.
Give the vertex of the graph of the function
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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 
.
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
or, alternatively written,
The graph of
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Solve the inequality:
Solve the inequality:
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The inequality compares an absolute value function with a negative integer. Since the absolute value of any real number is greater than or equal to 0, it can never be less than a negative number. Therefore, 
can never happen. There is no solution.
The inequality compares an absolute value function with a negative integer. Since the absolute value of any real number is greater than or equal to 0, it can never be less than a negative number. Therefore,
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Solve for 
:
Solve for
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Solve for positive values by ignoring the absolute value. Solve for negative values by switching the inequality and adding a negative sign to 7.
Solve for positive values by ignoring the absolute value. Solve for negative values by switching the inequality and adding a negative sign to 7.
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