The first step is to split the absolute value into two equations, one case for when the inside of the absolute value will be negative and one case for when the inside of the absolute value will be positive. 

Positive case:

First subtracting 2 and 4x from both sides:

dividing both sides by -2:

Negative case:

First move the negative to the other side:

Next, add 4x and subtract 2 from both sides:

Finally, divide both sides by 6 and reducing:

Thus the solutions are:

Absolute value symbols tell you to take whatever number is inside the absolute value and make it positive. This is important to remember for equations with variables like x because if the variable is inside the absolute value symbols, that means two values for the variable will give the right answer. You solve for both the positive and the negative of whatever number is on the other side of the equation, since the absolute value symbols will turn both positive and negative numbers into positive ones. For example, in this problem, the absolute value of 2x+4 is alone on one side of the equation with a variable (x) in it. Set 2x+4 equal to both 16 and -16 in two different equations removing the absolute value sign. 

You should have  and . Solve to get x alone.

Subtract 4 from both sides of the equation to get

 and

Divide by 2 on both sides of the equation to get

 and

If you want to check these answers plug them back into the original equation.

Now try the other solution.

Both solutions for x check out to give the answer 16.

This problem should be easy if you have a basic understanding of absolute values. Knowing that absolute values make whatever numbers inside them positive, ask yourself, what values for x will equal positive or negative 9? You can again set up two equations.

and 

These two equations are already solved for x which means we have our answers.

Double checking by plugging these back into the original equation...

Plugging in 9.

Now to try plugging in -9.

Both answers check out

This question is a bit of a trick. Absolute values always make the numbers in them positive. If an absolute value is left alone on one side of the equation and the other side of the equation is a negative number then there cannot be a possible solution.

Here is where checking your answer is important! Say you did not see the negative on the right side of the equation and thought you were supposed to solve the absolute value for both the positive and negative value of the other side

Remember this is the incorrect way to do this problem!

 and

and

Plugging these answers back into the equation you can see why this is wrong.

and

Neither of these answers is right. There is no solution.

The absolute value is alone on one side of the equation, which means it is ready to be set equal to the positive and negative value of the other side of the equation while removing the absolute value signs.

You should get

and

Distribute the 2 in each equation to both the x and the 4 to get

and

Subtract 8 from both sides

and

Divide by 2 to both sides to get the answers

and

Plug each answer back into the original.

and

Both answers check out correctly.

This question is slightly different because the x variable is in the absolute value and the absolute value is not alone on one side of the equation. The first step is to get the absolute value alone by moving everything else on that side of the equation to the other side.

In this problem, that positive 5 is in the way so move it by subtracting 5 from both sides of the equation.

Now, because the absolute value is by itself on one side of the equation you can set it equal to the positive and negative value of the other side. You should have

and

Get x alone by first adding 9 to both sides of each equation

and

Divide by 4 from both sides of each equation for your answers, making sure to reduce the fraction you get in the second answer to its lowest terms.

and

Plug back in the original to check

and

Both answers successfully check out

The absolute value is alone on one side of the equation and ready to be set equal to the positive and negative of the other side of the equation.

and

It is easier to remove the divide by 3 from the left side of the equation first by multiplying both sides of the equations by 3

and

Then add 8 to both sides to get x by itself

and

Plug back into the original to check as you always should.

and

Two correct answers!

If you look at this problem too quickly, you may see an absolute value with the other side being a negative and immediately think no solution. However, the absolute value is not alone on the left side. It still has that negative sign in front, which means you need to either multiply or divide it out from both sides before solving.

Dividing out the negative will get you

Now, this looks similar to other problems you have seen. Set the absolute value equal to the positive and negative value of the other side

and

Subtract 12 from both sides of each equation

and

Put back into the original to check your work.

and

Once again, the absolute value is not alone on one side of the equation so the first step is making that happen. When a number is in front of an absolute value it is understood as multiplication similar to when a number is in front of a parentheses. Step one is to divide 5 from both sides to get the absolute value alone on the left of the equation.

Now, since the absolute value is alone on one side, set the other side equal to both its positive and negative and drop the absolute value signs.

 and

Subtract 10 from both sides of each equation

and

Put each of these back in the original to double check your answer

and

These answers are both correct.

Again, you need to be careful looking at the absolute value on this question. Notice the denominator is not in the absolute value. You have to move the 4 in the denominator to the other side of the equation. To do this multiplyboth sides by 4. You will then get.

Now since the absolute value is alone on one side of the equation you can remove the absolute value symbols and set the equation equal to both the positive and negative of the other side, which is 12 and -12.

and

Subtract 2 from both sides of the equations

and

Then divide by 4 on both sides of the equations to get x alone

 and 

Finally make sure to reduce to the fraction to lowest terms or change to a mixed number depending on your teacher's preferences

 or and  or

Plug back into the original (I will just be using the fractions not the mixed numbers but it is the same result)

and

Both answers check out