College Algebra, Chapter 1, 1.1, Section 1.1, Problem 90

Solve the equation $\displaystyle \frac{a+1}{b} = \frac{a-1}{b} + \frac{b+1}{a}$ for $a$

$
\begin{equation}
\begin{aligned}
\frac{a+1}{b} &= \frac{a-1}{b} + \frac{b+1}{a} && \text{Subtract both sides by } \left( \frac{a-1}{b} \right)\\
\\
\frac{a+1}{b} - \frac{a-b}{b} &= \frac{a-1}{b} + \frac{b+1}{a} - \frac{a-1}{b} && \text{Simplify}\\
\\
\frac{\cancel{a}+1-\cancel{a}+1}{b} &= \frac{b+1}{a} && \text{Get the LCD and combine like terms}\\
\\
\frac{2}{b} &= \frac{b+1}{a} && \text{Multiply both sides by } (ab)\\
\\
a\cancel{b} & \left[ \frac{2}{\cancel{b}} = \frac{b+1}{\cancel{a}} \right] \cancel{a}b && \text{Cancel out like terms} \\
\\
2a &= b(b+1) && \text{Divide both sides by 2}\\
\\
\frac{\cancel{2}a}{\cancel{2}} &= \frac{b(b+1)}{2} && \text{Simplify}\\
\\
a &= \frac{b(b+1)}{2}
\end{aligned}
\end{equation}
$

In Tennyson's "Ulysses," which words and lines indicate that life is much more than mere breathing?

Words and lines from Tennyson's "Ulysses" that reflect how life is much more than mere breathing explain how a person's life has to be defined with purpose and meaning. 
 
In Tennyson's poem, Ulysses is faced with a defining choice. He can either stay in Ithaca with his wife and serve his kingdom or go back out to the seas to confront the challenges that await.  He finds himself poised between domestic predictability or the uncertain future of what lies ahead.  One indication of the choice he is going to make is when he says, "I will drink / Life to the lees." Ulysses sees life as only having purpose when it is fully embraced.  He does not see this purpose as merely breathing.  Ulysses lives his life with a "hungry heart." These reflect his views of life.  They reflect passion and vitality, a duty to fight the elements until the very end and never relent. Ulysses does not feel that life is about merely breathing, pathetically existing from one moment to another. Rather, he feels that an individual must possess a zeal for living.  Ulysses believes that people are active agents of their own destinies, authors of their own narratives.
 
Ulysses speaks of the dangers in not living life to its fullest capacity. He wonders "How dull it is to pause, to make an end, / To rust unburnish'd, not to shine in use!"  He considers it a waste of a life to merely exist.  Rather, he feels that people must see life as "yearning in desire / To follow knowledge like a sinking star."  Living life means understanding the world and one's place in it.  There is no defined end to this pursuit, as Ulysses says it should continue "Beyond the utmost bound of human thought." Ulysses does not believe that his journey has ended simply because he has returned home. The journey of life never ends.  Rather, he feels that his life, and all life, must be dedicated to the idea that "Some work of noble note, may yet to be done." There will always be more adventures to experience, more horizons to explore, more chapters to write in the book of one's life. Ulysses is not going to live in a banal way by merely breathing and existing.  Rather, he is insistent that his "purpose holds / To sail beyond the sunset, and the baths / Of all the western stars, until I die."  These words make clear that Ulysses defines life as living the journey and facing whatever awaits him on it.

 
In the poem's closing lines, Ulysses affirms how life is more than breathing. These thoughts define what he sees as the meaning to all existence:


We are not now that strength which in old days
Moved earth and heaven, that which we are, we are;
One equal temper of heroic hearts, 
Made weak by time and fate, but strong in will 
To strive, to seek, to find, and not to yield.

Ulysses will live his life fighting all adversaries. This includes the effects of time and age, and anything else that might dare to cross him.  Even though he might not succeed in all of these battles, he promises "to strive, to seek, to find, and not to yield."  It affirms how life is much more than simply existing.  It consists of the will to persevere through challenges, and to zealously anticipate what comes next.  Ulysses does not know what the future holds.  However, that is an integral part of what makes life worth living, making it much more than merely breathing.

How did the Industrial Revolution affect each region of the US?

In the Northeast and Mid-Atlantic, manufacturing in factories and on ports became large industries in which scores of economically disenfranchised people moved to the urban areas to work in factories for incredibly long hours and incredibly low pay. In the South, the invention of the cotton gin mechanized the production of cotton, which made the textile industry in the northeast large and more efficient in a capitalist sense. The invention of the cotton gin made the growing and production of cotton even more profitable and actually increased the demand for land and enslaved black people to toil in the fields and pick the cotton. Worker exploitation, child labor, horrific housing and labor conditions, and an ever increasing wealth gap became especially prominent in the Northeast and Mid-Atlantic regions. In the West, railroad companies brought industrialization westward as immigrant labor painstakingly built the railroads that further colonized the West and displaced indigenous peoples from their homelands.

To understand the effects of the industrial revolution on "each region," let's divide the nation into the four cardinal directions and consider major developments.
The South benefited from Eli Whitney's invention of the cotton gin, and cotton became a major crop as the South remained largely agrarian in this period, both before and after the Civil War. 
In the North, the industrial revolution created an economy based on manufacturing.  Eli Whitney's cotton gin and the development of interchangeable parts enabled the North to dominate manufacturing.
The East, particularly the Northeast, became the center of textile production during the early years of the industrial revolution; this is in part due to the extensive river system in this region that provided water power for the mills.
In the West, the completion of the transcontinental railroad by the joining of the Union Pacific and the Central Pacific lines enabled mass transit of both passengers and freight to move industrialization west. 
 
 
 

What were the major characteristics and leaders of the Indian nations living west of the Mississippi in 1860?

It is hard to categorize all the native Americans living west of the Mississippi in 1860 through a few characteristics.  The Plains Indians were heavily dependent on their horses and following the buffalo.  While they did winter in river valleys, most of the spring and summer grazing season was spent in warfare against other tribes and hunting the buffalo.  Indian warfare was not about how many people could be killed, but rather how brave one's fighters were.  It was considered honorable to touch your enemy in battle, and it was common for a group of young men to steal another tribe's horses.  
Around the Pacific Coast and Snake River in Idaho, fishing was common.  Many Indian groups around the Oregon Trail had already become dependent on forts and trading posts, and the gatherer tribes of northern and central California had been nearly driven to extinction due to the California gold rush and the Spanish mission movement.  The Indian of the American Southwest such as the Apache conducted raids against American mining camps, Mexican settlements across the border, and Navajo herdsmen.  The Navajo were mainly known for their sheep and their elaborate dress and artwork.  
Tribes did not have one leader who spoke for the whole nation, much to the consternation of the American government, who wanted to sign treaties with one group in order to get land rights.  Many tribes also had war chiefs and shamans, so the word "chief" can mean different things here.  Cochise was a leader of the Comanche Indians during the Red River War, and Sitting Bull and Crazy Horse were leaders of the Lakota Sioux.  Red Cloud also led the Lakota in the late 1860s in the effort to keep the settlers off the Bozeman Trail, and he is the only native leader to have a successful war against the U.S.  Geronimo was one of the last Apache leaders to surrender his small group, and Chief Joseph of the Nez Perce nearly escaped with his band to Canada in a chase that was chronicled by the major newspapers in the East.  

Beginning Algebra With Applications, Chapter 5, 5.4, Section 5.4, Problem 14

Using the slope intercept form, determine the equation of the line that contains the point whose coordinates are $(-1,2)$ and has slope of $\displaystyle \frac{-1}{2}$.


$
\begin{equation}
\begin{aligned}

y =& mx+b
&&
\\
y =& \frac{-1}{2} x + b
&& \text{Replace $m$ with } \frac{-1}{2}
\\
2 =& \frac{-1}{2} (-1) + b
&& \text{Substitute $x = -1$ and $y = 2$}
\\
2 =& \frac{1}{2} + b
&& \text{Solve for } b
\\
\\
\frac{3}{2} =& b
&& \text{Add } \frac{-1}{2}
\\
\\
y =& \frac{-1}{2} x + \frac{3}{2}
&& \text{Write the equation of the line by replacing $m$ and $b$ in $y = mx+ b$ by their values}

\end{aligned}
\end{equation}
$

Calculus of a Single Variable, Chapter 8, 8.2, Section 8.2, Problem 13

To evaluate the integral: int x^3e^x dx , we may apply "integration by parts": int u *dv = uv- int vdu .
Let: u= x^3 then du = 3x^2 dx
dv = e^x dx then v = int e^x dx = e^x .

Apply the formula for integration by parts, we get:
int x^3e^x dx = x^3 e^x - int 3x^2e^xdx .
= x^3 e^x - 3 int x^2e^xdx.
To evaluate int x^2 e^x dx , we apply another set of integration by parts.
Let: u = x^2 then du = 2x dx
v=e^x dx then dv = e^x
The integral becomes:
int x^2 e^x dx =x^2e^x - int 2xe^x dx
Another set of integration by parts by letting:
u = 2x then du =2dx
v=e^x dx then dv = e^x
int 2xe^x dx = 2xe^x - int 2e^x dx
= 2xe^x -2 e^x +C
Using int 2xe^x dx =2xe^x - 2e^x +C , we get:
int x^2 e^x dx =x^2e^x - int 2xe^x dx
=x^2e^x - [2xe^x - 2e^x ]+C
=x^2e^x - 2xe^x + 2e^x +C
Then,
int x^3e^x dx = x^3 e^x - 3 int x^2e^xdx .
= x^3 e^x - 3 [x^2e^x - 2xe^x + 2e^x] +C
= x^3 e^x - 3x^2e^x +6xe^x -6 e^x +C

Why do sociologists find it important to differentiate between sex and gender? What importance does the differentiation have in modern society?

I would add to the excellent points made by the first educator that the traditional understanding of "sex" did not allow for the fact that a significant minority of people—about 1% of the global population—are intersex. This means that they do not have the genitalia and reproductive equipment we identify as being either "male" or "female." Historically, it was common to try to turn all intersex babies into girls, but a good understanding of what gender is has allowed for advances in the way intersex people are treated. Rather than attempting to force intersex babies to identify as female, studies have found that the natural gender traits of intersex people will often exert themselves, regardless of how they were socialized as children, such that a child who has been raised as a girl may experience gender dysphoria as a teenager upon recognizing that he feels himself to be a boy. As such, there is now a strong argument for raising intersex babies without specific gender until they hit puberty and allowing these children, who are not defined by any specific binary "sex", to determine their own gender.
Conflating "sex" with "gender," then, is problematic because although many people Assigned Female At Birth (female-sexed) do grow up to feel that their gender expression is female, this is not always the case because sex is a physical trait, and gender is, indeed, a matter of self-expression. This idea becomes more obvious when we consider those people who do not have a defined sex; historically this caused panicked attempts to carve out a "gender" box for them, whereas in fact, we are now learning that our gender is something we can define for ourselves and need not correspond with any defined biological sex.

In sociology, sex refers to the biological characteristics that a society uses to categorize people as either male or female. For example, each of the sexes has specific genitalia. Other secondary characteristics that are acquired upon adolescence are the development of breasts and broadening of hips among girls and the deepening of voices among boys.
Gender, on the other hand, explains how societies identify and take care of the different sexes. This includes a cultural interpretation of the roles played by either sex in the society. It also looks at how the society influences our understanding of the sexes. It addresses what is seen as "male" or "female" behavior—aspects that are not defined by the male or female biology. It addresses questions such as "can a man wear a dress?" or "why do we have distinct names that separate boys from girls?”
The distinction between sex and gender is important, because sometimes a person’s sex may not align with his or her gender. There is pretty clear evidence that gender is socially constructed. Different societies can ascribe different behavior to a gender. For instance, while it is weird for men to wear dresses in certain societies, other societies in Africa, Asia, and the Middle East have a culture that encourages the wearing of dresses or even skirts among men.
In modern society, a distinction between sex and gender is also important because of changing gender roles, for instance in occupations. Women are increasingly taking up jobs that were previously considered "male jobs," (e.g., heavy truck driving, technology jobs or even politics). As such, the term "gender" rather than "sex" is used to remove traditional societal biases between the sexes.
https://www.monash.edu/medicine