The first step to solving is to use the Order of Operations.

\(\displaystyle \left |-8 + 3 (16) \right |\)

\(\displaystyle \left | -8 +48 \right |\)

The absolute value \(\displaystyle \left | x\right |\) of a real number x is the non-negative value of x without regard to its sign.  The absolute value of a number is the distance of that number from the point of origin or zero on a number line.

\(\displaystyle \left | 40\right | = 40\)

To solve this Absolute Value inequality, remove the Absolute Value bars and create two linear inequalities.

\(\displaystyle 3 (\frac{1}{2}x) + 6> 12\)

\(\displaystyle 3 (\frac{1}{2}x) + 6 < -12\)

Then, using the Order of Operations and the method for solving multi-step equations solve.

First inequality:

\(\displaystyle 3 (\frac{1}{2}x) + 6> 12\)

\(\displaystyle \frac{3}{2}x + 6 > 12\)

\(\displaystyle \frac{3}{2}x + 6 - 6 >12 -6\)

\(\displaystyle \frac{3}{2}x >6\)

\(\displaystyle \frac{3}{2}x (\frac{2}{3}) >\frac{6}{1} (\frac{2}{3})\)

\(\displaystyle x> 4\)

Second inequality:

\(\displaystyle \frac{3}{2}x +6 < -12\)

\(\displaystyle \frac{3}{2}x +6 -6 < -12 -6\)

\(\displaystyle \frac{3}{2}x < -18\)

\(\displaystyle \frac{3}{2}x (\frac{2}{3}) < -\frac{18}{1} (\frac{2}{3})\)

\(\displaystyle x< -12\)

The solution to  \(\displaystyle 3\left | \frac{1}{2}x +2 \right | > 12\) consists of the two intervals \(\displaystyle x > 4\) and  \(\displaystyle x < -12\).  This pair of inequalities is the solution.

To solve this Absolute Value inequality, remove the Absolute Value bars and create two linear inequalities.

\(\displaystyle 4x + 4 < 16\) and

\(\displaystyle 4x + 14 > -16\)

Then solve each of the inequalities.

First inequality:

\(\displaystyle 4x + 4 < 16\)

\(\displaystyle 4x +4 - 4 < 16-4\)

\(\displaystyle 4x < 12\)

\(\displaystyle \frac{4x}{4} < \frac{12}{4}\)

\(\displaystyle x < 3\)

Second inequality:

\(\displaystyle 4x + 14> -16\)

\(\displaystyle 4x + 4 -4 > -16-4\)

\(\displaystyle 4x > -20\)

\(\displaystyle \frac{4x}{4}> \frac{-20}{4}\)

\(\displaystyle x> -5\)

The solution to   consists of the two intervals \(\displaystyle x< 3\) and  \(\displaystyle x > -5\).  This pair of inequalities is the solution.

To solve this Absolute value inequality, remove the absolute value bars and create two linear inequalities and solve.

\(\displaystyle 7x\geq 28\)

\(\displaystyle x\geq4\)

\(\displaystyle 7x \leq -28\)

\(\displaystyle x \leq -4\)

The solution to \(\displaystyle \left | 7x\right | \geq 28\) consists of the two intervals \(\displaystyle x\geq4\) and  \(\displaystyle x \leq -4\).

This pair of inequalities is the solution.

Step One: Order of Operations

\(\displaystyle -\left |-8 - 2 x 3^{2} \right |\)

\(\displaystyle -\left | -8-2x9\right |\)

\(\displaystyle -\left | -8-18\right |\)

\(\displaystyle -\left | -26\right |\)

The absolute value of \(\displaystyle -26\) is \(\displaystyle 26.\)  However, because there is a negative sign outside the Absolute value bars, you would multiply the \(\displaystyle 26\) by \(\displaystyle -1\) to get the solution.

The correct answer is \(\displaystyle -26\)

\(\displaystyle \left | s-4000\right | \leq 600\) is the correct solution.

This absolute value inequality states that the difference between the salary of $4,000 must be less than or equal to $600 to be tolerable.

\(\displaystyle \left | h-145\right | \leq 5\) is the correct solution.

This Absolute Value inequality states that the difference between the children's height and \(\displaystyle 145cm\) must be less than or equal to \(\displaystyle 5cm\).

The absolute value of a number is its distance from zero. The absolute value of a negative integer is a positive integer. The distance from  zero to \(\displaystyle -9\) on a number line is \(\displaystyle 9\), so

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

The absolute value of a number is its distance from zero. The absolute value of a negative integer is a positive integer. The distance from 0 to \(\displaystyle -7\)on a number line     is \(\displaystyle 7\). Therefore:

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

Therefore

\(\displaystyle \left | -9\right |> \left | -7\right |\)

because

\(\displaystyle 9>7\)

\(\displaystyle \left | 4x+3y \right | - \left | 4x-3y \right |\)

To solve, we replace each variable with the given value.

\(\displaystyle \left | 4 (5) +3 ( 7) \right | - \left | 4 (5) -3 ( 7) \right |\ .\)

\(\displaystyle \left | 20 +21 \right | - \left | 20 -21 \right |\)

Simplify. Remember that terms inside of the absolute value are always positive.

\(\displaystyle \left | 41 \right | - \left | -1 \right |=41-1 = 40\)

 \(\displaystyle \begin{vmatrix}\; \; \left | 5- 3 \cdot 4\right |-8 \right \;\end{vmatrix}=\begin{vmatrix}\; \; \left | 5- 12\right |-8 \right \;\end{vmatrix}=\begin{vmatrix}\; \; \left | -7\right |-8 \right \;\end{vmatrix}=| 7 - 8| = | -1| = 1\)