$\displaystyle 7+\left | 5-9 \right |=7+\left | {-4} \|$

For an absolute value, the term inside the bars will always become positive.

$\displaystyle 7+\left | -4 \right |=7+4=11$

$\displaystyle 6-3\left | 4-6 \right |=6-3\left | -2 \right |$

For an absolute value, the term inside the bars will always become positive.

$\displaystyle 6-3\left | -2 \right |=6-3(2)=6-6=0$

$\displaystyle \small \small 6\left | 3-8 \right |+14$

Simplify the absolute value, as if it were a parenthesis.

$\displaystyle \small \small 6\left | -5 \right |+14$

The absolute value of $\displaystyle -5$ is $\displaystyle 5$. Remember that absolute values are always positive.

$\displaystyle \small (6)(5)+14$

$\displaystyle \small 30+14=44$

An absolute value is always the positive solution.

$\displaystyle \small \left | -7 \right |=7$

We can compare the terms by substituting the positive value of $\displaystyle 7$, and solving the right side of the inequality.

$\displaystyle \small \small \small \left | -7 \right |>12-6$

$\displaystyle \small \left | -7 \right |>6$

$\displaystyle \small 7>6$

The inequality is a true expression.

To solve problems with absolute value, first solve inside the absolute value signs. $\displaystyle -6+5-2 = -3$ .

The absolute value of $\displaystyle -3$ is $\displaystyle 3$ because the negative sign is inside the absolute value signs. Remember the absolute value is the distance from that number to zero on the number line. Therefore, the absolute value is always positive. 

First, treat the absolute value as a parenthesis, and evaluate the term inside.

$\displaystyle \left | 3-5\right |$

$\displaystyle \left | -2\right |$

The absolute value of any term is its distance from zero. A distance cannot be negative, thus, any negative term in an absolute value will be converted to a positive term.

$\displaystyle \left | -2\right |=2$

The absolute value of a number can be thought of as its distance from zero. $\displaystyle -3$ is three units away from zero. Therefore, $\displaystyle \left |-3 \right |=3$.

The absolute value of a number can be thought of as its distance from zero.

We start by solving the expression inside the absolute value signs and then measure how far that is from zero.

$\displaystyle \left | 3^2-12\right |$

$\displaystyle =\left | 9-12\right |$

$\displaystyle =\left | -3\right |$

$\displaystyle -3$ is three units away from zero. Therefore, $\displaystyle \left | 3^2-12\right |=\left |-3 \right |=3.$

The absolute value of a number can be thought of as its distance from zero.

We start by solving the expression inside the absolute value signs and then measure how far that is from zero.

$\displaystyle \left | -5+3\right |$

$\displaystyle =\left | -2\right |$

$\displaystyle -2$ is two units away from zero. Therefore $\displaystyle \left | -5+3\right |=\left | -2\right |=2.$

The absolute value of a number can be thought of as its distance from zero.

We start by solving the expression inside the absolute value signs and then measure how far that is from zero.

$\displaystyle \left | 12+24\right |$

$\displaystyle =\left | 36\right |$

$\displaystyle 36$ is $\displaystyle 36$ units away from zero. That means that $\displaystyle \left | 12+24\right |=\left | 36\right |=36.$