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

\(\displaystyle 7 < 9\)

The symbol | | around each real number signifies absolute value. Absolute value is the true magnitude of a real number, without looking at the positive or negative sign attached to it. To rewrite this problem when solving, remove all absolute value symbols as well as signs.

\(\displaystyle 15 + 24 = 39\)

The absolute value of a number is defined as the position or distance of the number from zero on the number.  Because it is a distance, the value cannot be negative.  So,

\(\displaystyle \left | -12\right | = 12\)

and

\(\displaystyle \left | -4\right | = 4\)

so

\(\displaystyle \left | -12\right | + \left | -4\right | = 12 + 4 = 16\)

Substitute 5 for \(\displaystyle x\) in the given equation and evaluate.

\(\displaystyle |-18 - |5+ 18 ||= |-18 - |23 ||\)

Remember that the absolute value of a number is its distance from zero on a number line. Distance is always positive; therefore, you can rewrite the expression.

\(\displaystyle |-18 - |23 ||= |-18 - 23|\)

Subtracting a positive number from a negative number is the same as adding a negative number. 

\(\displaystyle |-18 - 23|= |-18 +\left ( - 23 \right )|\)

\(\displaystyle |-18 +\left ( - 23 \right )|= |-\left (18 + 23 \right )|\)

\(\displaystyle |-\left (18 + 23 \right )|= |- 41|\)

Solve.

\(\displaystyle |- 41|= 41\)