This question is no solution because once you get the absolute value alone on one side the other side is negative. The first step towards getting the absolute value alone is adding 11 to both sides

Then, divide by -2 to both sides

Now, you have an equation where the absolute value is alone on one side and the other side is negative. This is impossible so there is no solution.

If you had not seen this and accidentally solved for both the positive and negative values of the other side of the equation you would have

and

Subtract 4 from both sides to get x alone

and or

Plugging the fraction answers in the original equation you would see that neither of these answers work

and

No answers work. There is no solution.

This question is another one that may trick people into putting no solution if they are not looking carefully at the problem. First, get the absolute value alone on one side of the equation. Only then should you look at the other side of the equation to see if it is negative and without a solution. This is not the case in this problem.

Get the absolute value alone by adding 9 to each side of the equation

Now that the absolute value is alone on one side, drop the absolute value symbols and set it equal to the positive and negative 7 of the other side of the equation. You should have

and

Subtract 1 from both sides

and

Divide by 6 on both sides to get x alone

and   After reducing the fraction

Plug both back into the original

and

Both answers check out.

Sometimes when you do enough absolute value problems with variables inside of them, you forget how to do ones without a variable inside. In this case, you are only going to end up with one answer because you can immediately simplify and eliminate the absolute value.

No absolute value after simplifying means you are only going to have one answer, .

To solve this absolute value equation, we must first understand absolute values.

An absolute value is the distance a number is from zero on the number line. Because it is a distance, it cannot equal a negative number. 

However, the value inside the absolute value CAN be negative (once the absolute value is applied, the answer becomes positive).

So, given the equation,

we will have to create 2 separate equations without the absolute values (we just want to find the value of x located inside the absolute values).  One that is equal to a positive answer, and one that is equal to a negative answer.

and

So,

Then,

Therefore,  or .  Either of these answers would make the absolute value equation true.

First isolate the absolute value sign.  We will need to split the absolute value sign into its positive and negative components.

Divide by three on both sides.

Simplify both sides.

Eliminate the absolute value and solve.

Add two on both sides.

This is one solution.  

To find the other solution, break up the absolute value sign and add a negative sign in front of the left quantity.  The equation becomes:

Divide by negative one on both sides to move the negative sign to the other side.

Add two on both sides.

The solutions to this absolute value are:  .

In order to solve, we will need to separate the absolute value in its positive and negative components:

Solve the first equation.  Add three on both sides.

This is the first solution.

Solve the second equation.  Divide by negative one on both sides to move the negative sign to the right side.

Add three on both sides.

Simplify both sides.

The solutions are:  

Be careful not to enclose these two answers using interval notation because this problem is not an inequality and does not have a range of solutions!

If , it follows that . It further follows that 

, 

and

.

If , then . It follows that 

, 

and

None of the choices are equivalent to this expression.

It follows that either  or ; we solve both separately.

The solution set is .

If , then , and 

.

Therefore,