Solve the terms inside the absolute value first.

\(\displaystyle \left |-6-6 \right |^{-2} = \left | -12\right |^{-2}\)

Simplify the absolute value.  Any term inside the absolute value is converted to a positive value.

\(\displaystyle (12)^{-2}\)

Convert the negative exponent to a positive exponent by taking the reciprocal of the inner term.  Simplify.

\(\displaystyle (12)^{-2} = \frac{1}{12^2} = \frac{1}{144}\)

The straight brackets indicate absolute value.

Absolute value means the distance from zero.

Example: Even a -16 is still 16 away from zero.

Therefore, \(\displaystyle \left | 3 * -5\right | + - 2 =\)

\(\displaystyle \left | - 15\right | + - 2 =\)

The absolute value of -15 is 15.

\(\displaystyle 15 + -2 = 13\)

The absolute value of a number is its distance from zero on a number line. Because of that, the absolute value of a number can never be negative.

So,

\(\displaystyle \left | -7\right | = 7\)

because the distance from zero to -7 on a number line is 7.

So,

\(\displaystyle \left | -7 \right | \cdot \left | 6\right | = 7 \cdot 6 = 42\)

First, we solve the absolute value portion:

\(\displaystyle \left | 8-15\right |\div -8\)

\(\displaystyle \left | -7\right |\div -8\)

\(\displaystyle 7\div -8\)

\(\displaystyle \frac{7}{-8}\)

\(\displaystyle -\frac{7}{8}\)

The term \(\displaystyle \left | -9\right |\) displays an absolute value.  Absolute value is defined as the distance of a number in relation to zero on a number line.  Since it is a distance, an absolute value cannot be negative.  So,

\(\displaystyle \left | -9\right | = 9\)

So, we can rewrite the orignal problem as

\(\displaystyle 9\ \square \ 4\)

and we can easily see that

\(\displaystyle 9 > 4\)

The absolute value of a number is the distance that number is from zero on a number.  Because it is a distance, the absolute value can never be negative.  So,

\(\displaystyle \left | -18\right | \square \left | 18\right |\)

can be written as

\(\displaystyle 18 \ \square \ 18\)

Therefore, it's easy to solve.

\(\displaystyle 18 = 18\)

Simplify the following expression:

\(\displaystyle \left | -44+38\right |\)

Let's begin by focusing on the inside of the absolute value signs:

\(\displaystyle -44+38=-6\)

However, because we are within the absolute value signs, we need to change the negative sign to a positive one.

\(\displaystyle \left | -6\right |=6\)

Anytime you have something within the absolute value sign, you need to make it positive.

Our answer is 6.

It is important to be careful of where the negative sign is when simplifying.

When simplifying you should end up with:

\(\displaystyle 4+4-2+12\)  which equals  \(\displaystyle 18\)

The absolute value is defined as the distance that number is from zero on the number line.  Because it is a distance, the absolute value cannot be negative. So, in the equation

\(\displaystyle 2x + \left | -9 \right | = 15\)

we look at the absolute value.

\(\displaystyle \left | -9\right | = 9\)

because -9 is 9 units from zero.  Now, we can solve for x.

\(\displaystyle 2x + 9 = 15\)

\(\displaystyle 2x + 9 - 9 = 15 - 9\)

\(\displaystyle 2x = 6\)

\(\displaystyle \frac{2x}{2} = \frac{6}{2}\)

\(\displaystyle x = 3\)

Absolute values are defined as the distance the number is from zero on a number line.  Because it is a distance, the absolute value cannot be negative.  So,

\(\displaystyle \left | -9 \right | = 9\)

because -9 is 9 units from zero on a number line.  So,

\(\displaystyle 7 \ \square \ 9\)

can easily be solved as

\(\displaystyle 7 < 9\)