We can rewrite this as an equation, where  is the number in question:

A nonnegative number is equal to its own absolute value, so if this number exists, it must be negative.

In thsi case, , and we can rewrite that equation as

This is the only number that fits the criterion.

Since , we know the following:  

 ;

;

;

;

.

Also, we need to compare absolute values, so the greatest one must be either  or .

We also know that  when .

Thus, we know for sure that .

We can rewrite this as an equation, where  is the number in question:

If  is nonnegative, then , and we can rewrite this as 

Solve:

If  is negative, then , and we can rewrite this as 

The numbers  have the given characteristics.

The correct answer is that there is no .

We start by adding  to both sides giving

Then multiply both sides by .

Then divide both sides by 

Now it is impossible to go any further. The absolute value of any quantity is always positive (or sometimes ). Here we have the absolute value of something equaling a negative number. That's never possible, hence there is no  that makes this a true equation.

We start by isolating the expression with the absolute value:

  becomes 

So:  or 

We then solve the two equations above, which gives us 42 and 4 respectively.

So the solution is 

We proceed as follows

(Start)

(Subtract 3 from both sides)

or (Quantity inside the absolute value can be positive or negative)

or  (add five to both sides)

or

Another way to say this is

To solve an inequality we need to remember what the absolute value sign says about our expression. In this case it says that

can be written as

 Of .

Rewriting this in one inequality we get:

From here we add one half to both sides .

Finally, we divide by two to isolate and solve for m.

Only  is between -1.75 and 2.25

This is a problem where we need to use our translating skills. We are given a word problem and asked to solve it. To do so, we need to rewrite our word problem as an equation and then use arithmetic to find the answer. In these types of problems, the hardest step is usually translating correctly, so make sure to be meticulous and work step-by-step!

1)"The absolute value of negative seventeen": Recall that absolute value means that we will just change the sign to positive. Missing that will end up giving you the trap answer .

2)"is multiplied by a number which is three fewer than twelve." We need a number that is three fewer than twelve, so we need to subtract. Follow it up with multiplication and you get:

3)"The resulting number is subtracted from negative six." The key word here is "from"—make sure you aren't computing , which would result in another one of the trap answers!

The correct answer is .

Since we are solving an absolute value equation, , we must solve for both potential values of the equation:

1.) 

2.) 

Solving Equation 1:

Solving Equation 2:

Therefore, for ,  or .

Since we are solving an absolute value equation, , we must solve for both potential values of the equation:

1.) 

2.) 

Solving Equation 1:

Solving Equation 2:

Therefore, for ,  or .