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.

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 .

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...

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:  

The statement shows that two numbers can be multiplied in either order to yield the same product. This is the commutative property of multiplication.

The statement demonstrates that two expressions both equal to a third expression are equal to each other. This is the transitive property of equality.

The idea that one multiplied by any number yields the latter number as the product is the identity property of multiplication.

Set up a proportion:

Cross-multiply:

Isolate by dividing by 8:

Finally, simplify:

The fraction is therefore equivalent to .

Set up a proportion:

Cross multiply:

Isolate by dividing both sides by 12:

Simplify:

(Hint: If you are unable to use long division to solve the fraction , take a look at your answer choices and see if you can narrow down your choices. You know that value is going to be greater than one, since 175 is greater than 3. You also know that dividing any number by 3 is going to result in a decimal of .33, .66, or no decimal at all. With these two pieces of information, you should be able to isolate the correct answer without actually solving for it like I did in the last step.)

To solve this proportion, cross multiply.

From here to solve, divide both sides by 121.