The absolute value will always yield a positive, as long it is not zero. Therefore, b cannot equal –7. For the value to be positive, a must be a negative number.

19 – 36(3) + 2(4 – 87) =

19 – 108 + 2(–83) =

19 – 108 – 166 = –255

Absolute value is the non-negative value of the expression

Absolute value problems generally have two answers:

z + 1 < 3 or z + 1 > –3 and subtracting 1 from each side gives z < 2 or z > –4 which bcomes –4 < z < 2

\(\displaystyle 2^{3}=8\)

\(\displaystyle 8-12=-4\) and \(\displaystyle \frac{-4}{2}=-2\)

It is important to know that the absolute value of something is always positive so the absolute value of \(\displaystyle -2\) is \(\displaystyle 2\)

2 is your answer.

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

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

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

\(\displaystyle = \left | -14 \right | + \left | -1 \right |\)

\(\displaystyle =14 + 1 = 15\)

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,

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

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

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

\(\displaystyle = \frac{9}{5} \div \frac{8}{5}\)

\(\displaystyle = \frac{9}{5} \times \frac{5}{8}\)

\(\displaystyle = \frac{9}{8}\)

\(\displaystyle =1 \frac{1}{8}\)