# Algebra II : Solving Absolute Value Equations

## Example Questions

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### Example Question #71 : Solving Absolute Value Equations

Solve.

or

and

and

or

or

or

Explanation:

Solve.

Step 1: Isolate the absolute value by subtracting  from both sides of the equation.

Step 2: Divide -1 from both sides of the equation in order to get rid of the negative sign in front of the absolute value.

Step 3: Because this is an inequality, this equation can be solved in two parts as shown below.

Note:   can be written as  or

or

Step 4: Solve both parts.

Distribute the .

Subtract  from both sides of the equation.

Divide both sides of the equation by .

Distribute the .

Subtract  from both sides of the equation.

Divide both sides of the equation by .

Step 5: Combine both parts using "or".

or

Solution:  or

### Example Question #72 : Solving Absolute Value Equations

Solve for values given the equation

and

and

and

and

Explanation:

Given:

When given an absolute value 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  for the same reason there are potentially two solutions, which are shown by  as an absolute value will always end up creating a positive result.

To simplify the absolute value we must look at each of these cases:

Just like a normal equation with one unknown we will simplify it by isolating  by itself. This is first done by subtracting  from both sides leaving:

Next  is divided from both sides leaving , as your final solution.

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

so this becomes:

and the absolute value of  is  so you end up with a true statement. Therefore   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:

Next  can be added to both sides, giving

dividing by  leaves:

Checking this solution is done just as you did for the previous solution obtained.

Given

substitute   in for

multiply  gives

so you obtain

and the absolute value of  is  thereby making this also a valid solution, therefore the two valid solutions are   and

### Example Question #73 : Solving Absolute Value Equations

Solve for

and

and

and

and

Explanation:

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:

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 #5 : Solving Absolute Value Equations

Solve for :

Explanation:

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 #74 : Solving Absolute Value Equations

Which values of  provide the full solution set for the inequality: