Start by isolating the absolute value.

Subtract seven on both sides.

Divide by three on both sides.

There is no value of  in an absolute value that will give a negative value since the absolute value will convert all numbers to positive.

The answer is:  

What is inside the absolute value brackets could be positive or negative.

Positive:

Negative:

Separate the absolute value and solve both the positive and negative components of the absolute value.

Solve the first equation.  Add  on both sides.

Add two on both sides.

Divide by five on both sides.

One of the solutions is  after substitution is valid.

Evaluate the second equation.  Distribute the negative sign in the binomial.

Subtract  on both sides.

Add two on both sides.

If we substitute this value back to the original equation, the equation becomes invalid.  

The answer is:  

Divide by two on both sides to isolate the absolute value.

The equation becomes:

Split this equation into its positive and negative components.

Solve the first equation.  Add three on both sides.

Divide by three on both sides.

The first solution is:  

Evaluate the second equation.  Divide by negative one on both sides.

Add three on both sides.

Divide by three on both sides.

The second solution is:  

The answers are:  

Isolate the absolute value by dividing both sides by negative three.

The equation becomes:  

Recall that an absolute value cannot be negative since the absolute value will change negative values to positive.

The answer is:  

Subtract  on both sides.

Evaluate the positive and negative components of the absolute value.  The two equations are:

Evaluate the first equation.  Add  on both sides.

There is no solution for the first equation.

Evaluate the second equation.  Simplify the left side by distributing negative one through the binomial.

Add  on both sides.

The equation becomes:

Add two on both sides.

Divide by six on both sides and reduce the fraction.  Verify that the answer will satisfy the original equation.

The answer is:  

Evaluate by adding six on both sides.

Divide by negative two on both sides.

The equation becomes:  

Recall that absolute values will convert all negative values to positive.  No matter what  would evaluate to be, it will never become negative half.

The answer is:   

Divide both sides by eight.

Split the absolute value into its positive and negative components.

Solve the first equation.  Multiply both sides by eight to eliminate the fraction.

Use distribution to simplify.

Add 24 on both sides, and then divide both sides by eight.

The first solution is .

Solve the second equation.  Divide by negative one on both sides.

The equation becomes:

Add three on both sides.  This is the same as adding  on both sides.

The answers are:  

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 

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:

 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 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 

adding 

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