To simplify, we must first simplify the absolute values.

Now, combine like terms:

The absolute value of a number is the points away from zero on a number line.

Since this is a countable value, you cannot count a negative number.

This makes all absolute values positive and also make the statement above false.

To solve for x we need to make two separate equations. Since it has absolute value bars around it we know that the inside can equal either 7 or -7 before the asolute value is applied.

Divide both sides by negative three.

Since the lone absolute value is not equal to a negative, we can continue with the problem. Split the equation into its positive and negative components.

Evaluate the first equation by subtracting one on both sides, and then dividing by two on both sides.

Evaluate the second equation by dividing a negative one on both sides.

Subtract one on both sides.

Divide by 2 on both sides.

The answers are:

Substitute in the expression:

Step 1: Evaluate ...

Step 2: Apply the minus sign inside the absolute value to the answer in Step 1...

Step 3: Define absolute value...

The absolute value of any value is always positive, unless there is an extra negation outside (sometimes)..

Step 4: Evaluate...

Rewrite the quadratic equation in standard form by subtracting from both sides:

Factor this as

where the squares represent two integers with sum and product 14. Through some trial and error, we find that and work:

By the Zero Product Principle, one of these factors must be equal to 0.

If then ;

if then .

The given equation has solution set , so we are looking for an absolute value equation with this set as well.

This equation can take the form

This can be rewritten as the compound equation

Adding to both sides of each equation, the solution set is

and

Setting these numbers equal in value to the desired solutions, we get the linear system

Adding and solving for :

Backsolving to find :

The desired absolute value equation is .

The lines on the outside of this problem indicate it is an absolute value problem. When solving with absolute value, remember that it is a measure of displacement from 0, meaning the answer will always be positive.

For this problem, that gives us a final answer of 10.

Here, we have to split the problem up into two parts:

and

Let's start with the first equation:

First, we can add to each side:

Now we divide by -6. Remember, when you divide by a negative, you flip the sign of the inequality:

Which we can reduce:

Now let's do the other part of the problem the same way:

To solve an absolute value inequality, first isolate the absolute value expression, which can be done here by subtracting 35 from both sides:

There is no need to go further. The absolute value of any number is always greater than or equal to 0, so, regardless of the value of ,

.

Therefore, the solution set is the set of all real numbers.