$\displaystyle \left | 2x -18 \right | + \left | 3x - 7 \right |$

$\displaystyle = \left | 2 \cdot 2 - 18 \right | + \left | 3 \cdot 2 - 7 \right |$

$\displaystyle = \left | 4 - 18 \right | + \left | 6 - 7 \right |$

$\displaystyle = \left | -14 \right | + \left | -1 \right |$

$\displaystyle =14 + 1 = 15$

Substitute 0.6 for $\displaystyle x$ :

$\displaystyle \left | 4x - 1.4 \right | + \left | x^{2} - 1 \right |$

$\displaystyle = \left | 4 \cdot 0.6 - 1.4 \right | + \left | 0.6^{2} - 1 \right |$

$\displaystyle = \left | 2.4 - 1.4 \right | + \left | 0.36 - 1 \right |$

$\displaystyle = \left | 1 \right | + \left | -0.64 \right |$

$\displaystyle =1 + 0.64$

$\displaystyle = 1.64$

Substitute $\displaystyle x = 0.6$.

$\displaystyle \left |0.5 x - 0.7 \right | - \left |0.6 x - 0.4 \right |$

$\displaystyle = \left |0.5 \cdot 0.6 - 0.7 \right | - \left |0.6 \cdot 0.6 - 0.4 \right |$

$\displaystyle = \left |0.3- 0.7 \right | - \left |0.36 - 0.4 \right |$

$\displaystyle = \left |-0.4 \right | - \left | - 0.04 \right |$

$\displaystyle = 0.4 - 0.04 = 0.36$

$\displaystyle | x + 7 |$ is the absolute value of $\displaystyle x+ 7$, which in turn is the sum of a number and  seven and a number. Therefore, $\displaystyle | x + 7 |$ can be written as "the absolute value of the sum of a number and seven". Since it is equal to $\displaystyle x - 3$, it is three less than the number, so the equation that corresponds to the sentence is 

"The absolute value of the sum of a number and seven is three less than the number."

$\displaystyle f(x) = |3x - |x^{2}- 7|\; |$

$\displaystyle f(2) = |3 \cdot 2 - |2^{2}- 7|\; |$

$\displaystyle = |3 \cdot 2 - |4- 7|\; |$

$\displaystyle = |3 \cdot 2 - |-3|\; |$

$\displaystyle = |3 \cdot 2 -3 |$

$\displaystyle = |6 -3 |$

$\displaystyle = | 3 |$

$\displaystyle = 3$

$\displaystyle a \blacktriangledown b= \frac{a+1}{\left | a\right |+ \left | b\right |}$, or, equivalently,

$\displaystyle a \blacktriangledown b=\left ( a+1 \right ) \div ( | a |+ | b | )$

$\displaystyle \frac{4}{5} \blacktriangledown \left (-\frac{4}{5} \right )=\left ( \frac{4}{5}+1 \right ) \div \left (\left| \frac{4}{5} \right |+ \left |- \frac{4}{5}\right | \right )$

$\displaystyle = \frac{9}{5} \div \left ( \frac{4}{5} +\frac{4}{5} \right )$

$\displaystyle = \frac{9}{5} \div \frac{8}{5}$

$\displaystyle = \frac{9}{5} \times \frac{5}{8}$

$\displaystyle = \frac{9}{8}$

$\displaystyle =1 \frac{1}{8}$

$\displaystyle p(x) = \frac{\left |x+2 \right |-1}{\left |x+1 \right |-2}$, or, equivalently,

$\displaystyle p(x) =\left ( \left |x+2 \right |-1 \right ) \div \left ( \left |x+1 \right |-2 \right )$

$\displaystyle p\left (-1 \frac{1}{5} \right )=\left ( \left |-1 \frac{1}{5}+2 \right |-1 \right ) \div \left ( \left |-1 \frac{1}{5}+1 \right |-2 \right )$

$\displaystyle =\left ( \left | \frac{4}{5} \right |-1 \right ) \div \left ( \left |- \frac{1}{5} \right |-2 \right )$

$\displaystyle =\left ( \frac{4}{5} -1 \right ) \div \left ( \frac{1}{5} -2 \right )$

$\displaystyle = - \frac{1}{5} \div \left ( - \frac{9}{5} \right )$

$\displaystyle = \frac{1}{5} \div \frac{9}{5}$

$\displaystyle = \frac{1}{5} \times \frac{5}{9}$

$\displaystyle = \frac{1} {9}$

$\displaystyle a \triangleright b = \left | a- \frac{1}{2}b\right | + \left | \frac{1}{2} a+b\right |$

$\displaystyle \frac{1}{3} \triangleright 4 = \left | \frac{1}{3} - \frac{1}{2} \cdot 4\right | + \left | \frac{1}{2} \cdot \frac{1}{3} +4\right |$

$\displaystyle = \left | \frac{1}{3} - 2\right | + \left | \frac{1}{6} +4\right |$

$\displaystyle = \left | -1 \frac{2}{3} \right | + \left | 4 \frac{1}{6} \right |$

$\displaystyle = 1 \frac{2}{3} + 4 \frac{1}{6}$

$\displaystyle = 5 \frac{5}{6}$

$\displaystyle g(x)= \left | \left | 1,000- \sqrt{x}\right | - x^{3} \right |$

$\displaystyle g(16)= \left | \left | 1,000- \sqrt{16}\right | - 16^{3} \right |$

$\displaystyle = \left | \left | 1,000- 4 \right | - 16^{3} \right |$

$\displaystyle = \left | \left | 996 \right | - 16^{3} \right |$

$\displaystyle = \left | 996 - 16^{3} \right |$

$\displaystyle = \left | 996 - 4,096 \right |$

$\displaystyle = \left | -3,100 \right |$

$\displaystyle =3,100$

$\displaystyle a \blacklozenge b = \left | 2a- 2b + 5 \right |$

$\displaystyle (-4) \blacklozenge (-7) = \left | 2 (-4)- 2(-7) + 5 \right |$

$\displaystyle = \left |-8- (-14) + 5 \right |$

$\displaystyle = \left |-8+14 + 5 \right |$

$\displaystyle = \left |11 \right |$

$\displaystyle = 11$