|x – 3| = 2 means that it can be separated into x – 3 = 2 and x – 3 = –2.

So both x = 5 and x = 1 work.

x – 3 = 2 Add 3 to both sides to get x = 5

x – 3 = –2 Add 3 to both sides to get x = 1

Because of the absolute value signs,

or

Subtract 2 from both sides of both equations:

or

or

There are two answers to this problem:

and

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:

The absolute value of a number can never be a negative number. Therefore, no value of can make a true statement.

When dealing with absolute value, we need to consider positive and negative values.

Therefore, we will create two separate equations to solve.

Equation 1:

Multiply on both sides. .

Equation 2:

Multiply on both sides and divide on both sides. .

Therefore, the solutions are, .

When dealing with absolute value, we need to consider positive and negative values.

Therefore, we will create two separate equations to solve

and

The two negatives become positive for the first equation.

For the second equation divide both sides by to get .

Therefore, the solutions are

.

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.

To solve for we set up the two equations.

and

For the first equation

And for the second equation

And so the solutions are

and