This document discusses how to solve absolute value equations and inequalities. It explains that absolute value equations have solution sets that include both the positive and negative values that satisfy the equation. Absolute value inequalities can be solved using the properties that absolute values are less than a number if the value is between the negatives of the number, and greater than a number if it is less than the negative or greater than the positive of the number. The document provides examples of solving absolute value equations and inequalities and discusses special cases when the absolute value expression is always true, false, or can be treated as a normal equation.
* Solve equations involving rational exponents
* Solve equations using factoring
* Solve equations with radicals and check the solutions
* Solve absolute value equations
* Solve other types of equations
* Solve polynomial equations by factoring
* Solve equations with radicals and check the solutions
* Solve equations with rational exponents
* Solve equations that are quadratic in form
* Solve absolute value equations
* Solve equations in one variable algebraically.
* Solve a rational equation.
* Find a linear equation.
* Given the equations of two lines, determine whether their graphs are parallel or perpendicular.
* Write the equation of a line parallel or perpendicular to a given line.
Identify basic properties of equations
Solve linear equations
Identify identities, conditional equations, and contradictions
Solve for a specific variable (literal equations)
* Solve a system of nonlinear equations using substitution.
* Solve a system of nonlinear equations using elimination.
* Graph a nonlinear inequality.
* Graph a system of nonlinear inequalities.
* Solve quadratic equations by factoring.
* Solve quadratic equations by the square root property.
* Solve quadratic equations by completing the square.
* Solve quadratic equations by using the quadratic formula.
* Identify the degree and leading coefficient of polynomials.
* Add and subtract polynomials.
* Multiply polynomials.
* Identify special product patterns.
* Perform operations with polynomials of several variables.
* Model exponential growth and decay
* Use Newton's Law of Cooling
* Use logistic-growth models
* Choose an appropriate model for data
* Express an exponential model in base e
* Construct perpendicular and angle bisectors
* Use bisectors to solve problems
* Identify the circumcenter and incenter of a triangle
* Use triangle segments to solve problems
* Identify, write, and analyze conditional statements
* Write the inverse, converse, and contrapositive of a conditional statement
* Write a counterexample to a fake conjecture
* Find the distance between two points
* Find the midpoint of two given points
* Find the coordinates of an endpoint given one endpoint and a midpoint
* Find the coordinates of a point a fractional distance from one end of a segment
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
* Introduce functions and function notation
* Develop skills in constructing and interpreting the graphs of functions
* Learn to apply this knowledge in a variety of situations
* Recognize graphs of common functions.
* Graph functions using vertical and horizontal shifts.
* Graph functions using reflections about the x-axis and the y-axis.
* Graph functions using compressions and stretches.
* Combine transformations.
* Identify intervals on which a function increases, decreases, or is constant
* Use graphs to locate relative maxima or minima
* Test for symmetry
* Identify even or odd functions and recognize their symmetries
* Understand and use piecewise functions
* Determine whether a relation or an equation represents a function.
* Evaluate a function.
* Use the vertical line test to identify functions.
* Identify the domain and range of a function from its graph
* Identify intercepts from a function’s graph
* Solve counting problems using the Addition Principle.
* Solve counting problems using the Multiplication Principle.
* Solve counting problems using permutations involving n distinct objects.
* Solve counting problems using combinations.
* Find the number of subsets of a given set.
* Solve counting problems using permutations involving n non-distinct objects.
* Use summation notation.
* Use the formula for the sum of the first n terms of an arithmetic series.
* Use the formula for the sum of the first n terms of a geometric series.
* Use the formula for the sum of an infinite geometric series.
* Solve annuity problems.
* Find the common ratio for a geometric sequence.
* List the terms of a geometric sequence.
* Use a recursive formula for a geometric sequence.
* Use an explicit formula for a geometric sequence.
* Find the common difference for an arithmetic sequence.
* Write terms of an arithmetic sequence.
* Use a recursive formula for an arithmetic sequence.
* Use an explicit formula for an arithmetic sequence.
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The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
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3. Absolute Value Equations
⚫ You should recall that the absolute value of a number a,
written |a|, gives the distance from a to 0 on a number
line.
⚫ By this definition, the equation |x| = 3 can be solved by
finding all real numbers at a distance of 3 units from 0.
Both of the numbers 3 and ‒3 satisfy this equation, so
the solution set is {‒3, 3}.
4. Absolute Value Equations (cont.)
⚫ The solution set for the equation must include
both a and –a.
⚫ Example: Solve
=x a
− =9 4 7x
5. Absolute Value Equations (cont.)
⚫ The solution set for the equation must include
both a and –a.
⚫ Example: Solve
The solution set is
=x a
− =9 4 7x
− =9 4 7x − = −9 4 7x
− = −4 2x − = −4 16x
=
1
2
x = 4x
or
1
,4
2
6. Absolute Value Inequalities
⚫ For absolute value inequalities, we make use of the
following two properties:
⚫ |a| < b if and only if –b < a < b.
⚫ |a| > b if and only if a < –b or a > b.
⚫ Example: Solve − + 5 8 6 14x
7. Absolute Value Inequalities (cont.)
⚫ Example: Solve
The solution set is
− + 5 8 6 14x
or
− 5 8 8x
− −5 8 8x − 5 8 8x
− −8 13x − 8 3x
13
8
x −
3
8
x
− −
3 13
, ,
8 8
8. Special Cases
⚫ Since an absolute value expression is always
nonnegative:
⚫ Expressions such as |2 – 5x| > –4 are always true. Its
solution set includes all real numbers, that is, (–, ).
⚫ Expressions such as |4x – 7| < –3 are always false—
that is, it has no solution.
⚫ The absolute value of 0 is equal to 0, so you can solve
it as a regular equation.
9. Classwork
⚫ 1.8 Assignment (College Algebra)
⚫ Page 163: 10-22 (even), page 155: 36-52 (4),
page 144: 72-88 (4)
⚫ 1.8 Classwork Check
⚫ Quiz 1.7
Your six weeks test is next Friday.
