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.

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

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  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   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,

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.

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 . 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 .

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: