Intermediate Algebra, Chapter 2, 2.7, Section 2.7, Problem 50

Solve $|-3 + x| > 8$. Graph the solution set.

The absolute value inequality is rewritten as

$-3 + x > 8$ or $-3 + x < -8$,

because $-3 + x$ must represent a number that is more than $8$ units from on either side of the number line.We can solve the compound inequality.


$
\begin{equation}
\begin{aligned}

-3 + x > & 8 && \text{or} &&& -3 + x < & -8
&&
\\
x > & 11 && \text{or} &&& x < & -5
&& \text{Add } 3

\end{aligned}
\end{equation}
$



The solution set is $(- \infty, -5) \cup (11, \infty)$.

What happens in an aqueous solution as the concentration of H3O+ decreases?

Each water molecule can form four hydrogen bonds with other water molecules due to its polar nature.
Therefore, a hydrogen atom taking part in a hydrogen bond between two water molecules can shift from one water molecule to the other. When this occurs, it leaves its electron behind. Therefore, the hydrogen ion will be a proton written as H+. The molecule that has lost its proton is called a hydroxide ion with the symbol OH-. The molecule that has accepted the extra proton is now called H30+ and is known as a hydronium ion. It is sometimes written as simply H+.
The process described above is dissociation of water molecules. However, these molecules can re-form water exists in a state of equilibrium with a neutral pH of 7.
Anytime a substance increases the H+ ion concentration in solution, it is called an acid. A base reduces the H+ ion concentration of a solution. Water is technically both an acid and a base. A shift in the number of H+ ions where they outnumbered the OH- ions would result in a lowered pH and the solution would be acidic. A shift in the number of H+ ions where they were less than the OH- ions would result in an increase in pH.
In a water- based solution at 25 degrees C, the product of H+ ions and OH- ions = 10 -14. Basically, in a neutral solution, there is an equal number of H+ and OH- ions.
The pH of a solution = -log (H+) therefore, in neutral water, the H ion concentration is 10-7 and if one uses the formula above, pH= - (-7)= 7. 
The pH scale is written from 1 to 14. Neutral water has a pH of 7 and values below 7 are increasingly stronger acids. Values above 7 and continuing to 14 are increasingly stronger bases.
In a solution, if the number of H30+ ions (also written as H+ ions) decreases, the pH of the solution will increase. There will be a shift toward more OH- negative ions as the H3O + ions decrease. 
 
 
https://www.sciencebuddies.org/science-fair-projects/references/acids-bases-the-ph-scale

H3O+ is called the hydronium ion in the aqueous solution. The variation of H3O+ in an aqueous solution affects the acidity of the said solution. Water molecules can dissociate into H3O+ and OH- at a given temperature. For example, at 25 degrees Celsius, H2O dissociates into:
2 H_2O = H_3 O^(+) + OH^(-)
With
K_w = [H_3 O^(+)] [OH^(-)]
Where:
[H_3 O^(+)]= Concentration of hydronium.
[OH^(-)]=Concentration of hydroxide.
Kw = 1.00x10^(-14), (Constant). 
Changes in the concentration of H3O+ in the solution are affected by the temperature and the species that is dissolved in water. 
Acidity is measured by the concentration of H3O+. It is represented as:
pH = -log [H_3 O^(+)] .
As the concentration of H3O+ decreases, pH increases. In other words, as the concentration of H3O+ decreases, the solution becomes less acidic.
In a neutral aqueous solution (pH = 7), there is roughly equal populations of H3O+ and OH-. Changing the pH or acidity of the solution can also mean changing the concentrations of H3O+ and OH-. 
 

H3O+ represents an acid.
As you might have guessed, H3O+ is nothing but H2O + H+
(H+ is a free positive ion, and in aqueous solution is will be coordinated by the H2O molecules, and hence H+ exists as H3O+)
H2O + H+ = H3O+
So, decreasing the H3O+ concentration implies a decrease in H+ concentration. So acidity of the solution decreases. Note: pH is a measure of the acidity of the solution. pH = - log [H3O+].
Thus, as H3O+ concentration decreases, pH increases.

Precalculus, Chapter 6, 6.3, Section 6.3, Problem 46

For a nonzero vector w a unit vector a with the same direction may be found by the formula
a=(1/||w||)*w.
It has the same direction because it is a w multiplied by a positive number.
It has a magnitude of 1 because
||a||=|1/||w|| |*||w||=(1/||w||)*||w||=1.

In this case ||w||=6 and a=-i. No doubt that ||a||=sqrt((-i)*(-i))=1.

What skills do business shareholders need?

A shareholder is anyone who has complete or partial ownership of a company. Someone who owns one share of a multi-billion dollar company as part of a modest investment portfolio is a shareholder, as is someone with a controlling interest in a technology startup. Perhaps the easiest way to think through the question of skills is to look separately at passive versus active shareholders.
A passive shareholder buys shares in a company as an investment but takes no part in actively running the company. This sort of shareholder needs a solid understanding of what makes a good investment but does not need detailed internal knowledge of how the company works. Often choice of shares is as much about balancing a portfolio or mirroring an index as about looking at individual businesses in detail. The basic skills required are some understanding of the nature of investing and finance, basic economic knowledge, patience, some mathematical skills, and discipline (avoiding emotional buying and selling). 
An active shareholder needs more detailed knowledge of the specific industry and a strong background in accounting and finance to understand what measures will increase the company's profitability.

What was the ritual that Jonas's family did at the end of their evening meals?

At the end of evening meals in The Giver, each family member is expected to express their feelings and thoughts about what happened that day. When viewed in a positive way, this gives each community member a chance to talk about their day. It is a time to hear about what is going on with the family and talk about their own experiences. When seen in a more negative light, this sharing can be seen as a way for parents to "spy" on their children. Having children share every thought they have gives parents the chance to learn about anything that may be concerning to the parents or the Elders. Jonas, Lily and their parents all share apprehensions or negative feelings during this time. This can be seen as both a way to bring the family closer and as a way for the Elders and parents to have control over members of the community.

In The Giver, families must follow certain rules and rituals. As an example, each night after dinner, Jonas and his family must participate in the "evening telling of feelings." Each family member takes a turn where he or she must use precise language to communicate thoughts and feelings. Other family members attempt to offer help or advice. In chapter 1, Lily is the first to share her feelings about her day. She describes her feelings of anger when a boy from another community doesn't follow the rules. Jonas's father is next, and he shares his concern about a "newchild." Next is Jonas's mother who describes her feelings in dealing with a repeat offender at the Department of Justice. Finally, Jonas shares that he is "apprehensive" regarding his upcoming Ceremony of Twelve.

What are common tactics used by interest groups to advance their agendas?

Interest groups try to advance their agendas by influencing decision makers.  They identify people or organizations that have the power to give them what they want and then try to make those people listen to them.  They use logic, data, and personal stories to try to get the decision maker(s) to identify with and care about their issue.    But, often a good argument is not enough.  Issue groups often try to find out what motivates people and use that motivation to get the change they seek.  For example, if an interest group is trying to influence a politician, they will try to convince the politician’s constituents to care about their issue and to let the politician know. They will also try to get coverage in the media. 

Special interest groups have many ways to accomplish their agendas. They can organize mass support for and only choose to support candidates for office who support their agendas. They can also lobby politicians directly in order to make their message heard. Another way that special interest groups can get their message out is through advertising.  
One example of how this worked in history was the work of Wayne Wheeler of the Anti-Saloon League. Wheeler organized industrialists and church groups to give money to support Prohibition. These people voted in blocs for candidates who would support the Volstead Act. Even if a candidate personally drank, he voted for the Volstead Act to avoid the wrath of Wayne Wheeler's special interest group. The Anti-Saloon League also spent money on advertising, including campaigns during World War I that tied alcohol to the unpatriotic waste of grain and the beer industry to German interests. While there are more contemporary interest groups, the Anti-Saloon League was one of the more powerful of its time. 

College Algebra, Chapter 5, 5.3, Section 5.3, Problem 12

Evaluate the expression $\displaystyle \log_{12} 9 + \log_{12} 16$


$
\begin{equation}
\begin{aligned}

\log_{12} 9 + \log_{12} 16 =& \log_{12} (9 \cdot 16)
&& \text{Law of Logarithm } \log_a (AB) = \log_a A + \log_a B
\\
\\
\log_{12} 9 + \log_{12} 16 =& \log_{12} (144)
&& \text{Simplify}
\\
\\
\log_{12} 9 + \log_{12} 16 =& 2
&& \text{Because } 12^2 = 144

\end{aligned}
\end{equation}
$