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Questions and Answers
What is the general form of a linear equation with one variable?
What is the general form of a linear equation with one variable?
Which of the following is an example of a quadratic equation?
Which of the following is an example of a quadratic equation?
What is the purpose of the multiplication and division properties in solving equations?
What is the purpose of the multiplication and division properties in solving equations?
What is the difference between a linear equation and a linear inequality?
What is the difference between a linear equation and a linear inequality?
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How do you solve a compound inequality?
How do you solve a compound inequality?
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What is the quadratic formula used for?
What is the quadratic formula used for?
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What is the purpose of factoring in solving quadratic equations?
What is the purpose of factoring in solving quadratic equations?
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What happens to the inequality sign when you multiply or divide both sides of a linear inequality by a negative coefficient?
What happens to the inequality sign when you multiply or divide both sides of a linear inequality by a negative coefficient?
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What was the main reason behind the Proclamation of 1763?
What was the main reason behind the Proclamation of 1763?
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What was the main goal of the American Patriots during the American Revolution?
What was the main goal of the American Patriots during the American Revolution?
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What was the outcome of the Mexican-American War?
What was the outcome of the Mexican-American War?
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What was the main issue that led to the American Civil War?
What was the main issue that led to the American Civil War?
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What was the main goal of the Reconstruction Era?
What was the main goal of the Reconstruction Era?
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What was the significance of the Louisiana Purchase?
What was the significance of the Louisiana Purchase?
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What was the main characteristic of the Robber Barons?
What was the main characteristic of the Robber Barons?
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What was the main outcome of the Spanish-American War?
What was the main outcome of the Spanish-American War?
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Study Notes
Equations
- An equation is a statement that says two expressions are equal
- It contains an equal sign (=) and is written in the form:
ax + by = c, wherea,b, andcare constants, andxandyare variables - Types of equations:
- Simple equations: have one variable and one solution (e.g.,
2x = 6) - Linear equations: have one variable and can be written in the form
ax + b = c, whereais not equal to zero (e.g.,2x + 3 = 5) - Quadratic equations: have one variable and can be written in the form
ax^2 + bx + c = 0, whereais not equal to zero (e.g.,x^2 + 4x + 4 = 0) - Systems of equations: consist of two or more equations with two or more variables (e.g.,
2x + 3y = 5andx - 2y = -3)
- Simple equations: have one variable and one solution (e.g.,
Solving Equations
- Addition and subtraction properties:
- If
a = b, thena + c = b + canda - c = b - c
- If
- Multiplication and division properties:
- If
a = b, thena × c = b × canda ÷ c = b ÷ c(ifcis not equal to zero)
- If
- Solving linear equations:
- Add or subtract the same value to both sides to isolate the variable
- Multiply or divide both sides by a coefficient to isolate the variable
- Solving quadratic equations:
- Factoring: express the equation as a product of binomials and set each factor equal to zero
- Quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Inequalities
- An inequality is a statement that says one expression is greater than, less than, or equal to another expression
- It contains an inequality sign (<, >, ≤, or ≥) and is written in the form:
ax + by > c, wherea,b, andcare constants, andxandyare variables - Types of inequalities:
- Linear inequalities: have one variable and can be written in the form
ax + b > c, whereais not equal to zero - Compound inequalities: consist of two or more inequalities with the same variable
- Linear inequalities: have one variable and can be written in the form
Solving Inequalities
- Addition and subtraction properties:
- If
a > b, thena + c > b + canda - c > b - c
- If
- Multiplication and division properties:
- If
a > b, thena × c > b × cifcis positive, anda × c < b × cifcis negative - If
a > b, thena ÷ c > b ÷ cifcis positive, anda ÷ c < b ÷ cifcis negative
- If
- Solving linear inequalities:
- Add or subtract the same value to both sides to isolate the variable
- Multiply or divide both sides by a coefficient to isolate the variable, and flip the inequality sign if the coefficient is negative
- Solving compound inequalities:
- Solve each inequality separately and combine the solutions using "and" or "or"
Equations
- An equation is a statement that says two expressions are equal, and is written in the form
ax + by = c, wherea,b, andcare constants, andxandyare variables. - Equations can be classified into simple, linear, quadratic, and systems of equations.
- Simple equations have one variable and one solution, such as
2x = 6. - Linear equations have one variable and can be written in the form
ax + b = c, whereais not equal to zero, such as2x + 3 = 5. - Quadratic equations have one variable and can be written in the form
ax^2 + bx + c = 0, whereais not equal to zero, such asx^2 + 4x + 4 = 0. - Systems of equations consist of two or more equations with two or more variables, such as
2x + 3y = 5andx - 2y = -3.
Solving Equations
- The addition and subtraction properties of equality state that if
a = b, thena + c = b + canda - c = b - c. - The multiplication and division properties of equality state that if
a = b, thena × c = b × canda ÷ c = b ÷ c(ifcis not equal to zero). - To solve linear equations, add or subtract the same value to both sides to isolate the variable, and multiply or divide both sides by a coefficient to isolate the variable.
- To solve quadratic equations, use factoring or the quadratic formula
x = (-b ± √(b^2 - 4ac)) / 2a.
Inequalities
- An inequality is a statement that says one expression is greater than, less than, or equal to another expression, and is written in the form
ax + by > c, wherea,b, andcare constants, andxandyare variables. - Inequalities can be classified into linear and compound inequalities.
- Linear inequalities have one variable and can be written in the form
ax + b > c, whereais not equal to zero. - Compound inequalities consist of two or more inequalities with the same variable.
Solving Inequalities
- The addition and subtraction properties of inequality state that if
a > b, thena + c > b + canda - c > b - c. - The multiplication and division properties of inequality state that if
a > b, thena × c > b × cifcis positive, anda × c < b × cifcis negative, and ifa > b, thena ÷ c > b ÷ cifcis positive, anda ÷ c < b ÷ cifcis negative. - To solve linear inequalities, add or subtract the same value to both sides to isolate the variable, and multiply or divide both sides by a coefficient to isolate the variable, and flip the inequality sign if the coefficient is negative.
- To solve compound inequalities, solve each inequality separately and combine the solutions using "and" or "or".
American History
Colonial Era (1607-1776)
- Establishment of Thirteen Colonies: British settlers founded colonies along the Eastern seaboard, with Virginia (1607) being the first.
- Pilgrims and Puritans: Pilgrims, seeking religious freedom, founded Plymouth Colony (1620), while Puritans, seeking to reform the Church of England, established the Massachusetts Bay Colony (1629).
- Triangular Trade: This system involved the exchange of slaves, goods, and resources between Europe, Africa, and the Americas, with the Middle Passage being the brutal journey of enslaved Africans to the Americas.
- Proclamation of 1763: The British decree limited westward expansion, leading to colonial discontent, as it prohibited settlement beyond the Appalachian Mountains.
American Revolution (1775-1783)
- Causes of the Revolution: Taxation without representation, restrictions on trade and commerce, and the slogan "No taxation without representation" fueled the rebellion.
- Key Events: The Boston Massacre (1770) marked the first bloodshed, the Boston Tea Party (1773) protested British taxation, and the Declaration of Independence (1776) formally declared American independence.
- American Patriots: Led by George Washington, they fought against British rule, with the Continental Army being formed in 1775.
- Treaty of Paris (1783): Recognized American independence from Great Britain, ending the Revolutionary War.
Westward Expansion and Civil War (1783-1865)
- Manifest Destiny: This belief in the United States' divine right to expand its territory led to westward expansion, fueled by a sense of nationalism and the idea of American exceptionalism.
- Louisiana Purchase (1803): The U.S. acquired approximately 828,000 square miles of land from France, doubling its territory.
- Mexican-American War (1846-1848): The U.S. gained a significant amount of land, including present-day California, Nevada, Utah, Arizona, New Mexico, Texas, and parts of Colorado, Kansas, Oklahoma, and Wyoming.
- American Civil War (1861-1865): The conflict between the Union (North) and Confederacy (South) revolved around slavery and states' rights, with the Emancipation Proclamation (1863) freeing all slaves in Confederate territory.
- Abraham Lincoln: The 16th President issued the Emancipation Proclamation, led the Union to victory, and ultimately abolished slavery with the 13th Amendment (1865).
Industrialization and Imperialism (1865-1914)
- Reconstruction Era (1865-1877): The period aimed to rebuild the South, establish civil rights for freed slaves, and redefine the nation's social and political landscape.
- Industrialization: Rapid growth of industry, technological innovation, and urbanization transformed the U.S. economy, with the development of railroads, steel, and oil industries.
- Robber Barons: Wealthy industrialists, such as John D. Rockefeller and Andrew Carnegie, dominated the economy, often using exploitative practices.
- Spanish-American War (1898): The U.S. emerged as a global power, gaining Puerto Rico, Guam, and the Philippines, and establishing its influence in the Caribbean and Pacific.
World War I and the Roaring Twenties (1914-1929)
- U.S. involvement in World War I (1917-1918): The U.S. entered the war on the Allied side, contributing to the defeat of the Central Powers, and suffering over 116,000 military casualties.
- The Roaring Twenties: Cultural, social, and economic changes defined the decade, with the Jazz Age, Prohibition, and the rise of consumer culture.
- Harlem Renaissance: This cultural and artistic movement among African Americans in the 1920s saw the emergence of notable figures like Langston Hughes, Zora Neale Hurston, and Duke Ellington.
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Description
Understand the concept of equations, including types of equations such as simple, linear, and quadratic equations. Learn how to write and solve equations in the form ax + by = c.
