The document provides an interview summary of a lesson on solving various types of inequalities:
1) The lesson covers linear, quadratic, and rational inequalities, explaining the steps to solve each type.
2) Examples are worked through demonstrating how to identify intervals on the number line and use test values to determine the solution set of inequalities.
3) Applications involving break-even points and projectile motion are presented to show real-world examples.
Objective Standard Setting_An application of Many Facet Rasch ModelSaidfudin Mas'udi
We often encounter multiple raters in an assessment. Rasch Model enable verification of a given data set that involved multiple assessors / raters. This is very useful in education where multiple Lecturers is involved in an assessment. Rasch ensure justice is done during the assessment on a student or a staff.
Objective Standard Setting_An application of Many Facet Rasch ModelSaidfudin Mas'udi
We often encounter multiple raters in an assessment. Rasch Model enable verification of a given data set that involved multiple assessors / raters. This is very useful in education where multiple Lecturers is involved in an assessment. Rasch ensure justice is done during the assessment on a student or a staff.
Special webinar on tips for perfect score in sat mathCareerGOD
Math is the language of logic and is therefore tested in all the major examinations where SAT is no exception.
Scoring well in Math can do wonders to your career and college candidature. Conversely, any complacency in Math affect your score and thus prove dangerous.
In this webinar, “Tips for perfect Math score in SAT and SAT- Math subject test” from the 5-day webinar series ‘Experts’ Speak: Demystifying US Admissions’, seasoned math trainers and subject experts with decades of experience in the industry share important insights on maximising your Math scores and minimising mistakes to lose out on Math scores.
Visit www.careergod.com for more info.
NUMERICA METHODS 1 final touch summary for test 1musadoto
MY FINAL TOUCH SUMMARY FOR TEST 1
ON 6TH MAY 2018
TOPICS AND MATERIALS COVERED
1. Class lecture notes (Basic concepts, errors and roots of function).
2. Lecture’s examples.
3. Past Years Examples.
4. Past Years examination papers.
5. Tutorial Questions.
6. Reference Books + web.
Special webinar on tips for perfect score in sat mathCareerGOD
Math is the language of logic and is therefore tested in all the major examinations where SAT is no exception.
Scoring well in Math can do wonders to your career and college candidature. Conversely, any complacency in Math affect your score and thus prove dangerous.
In this webinar, “Tips for perfect Math score in SAT and SAT- Math subject test” from the 5-day webinar series ‘Experts’ Speak: Demystifying US Admissions’, seasoned math trainers and subject experts with decades of experience in the industry share important insights on maximising your Math scores and minimising mistakes to lose out on Math scores.
Visit www.careergod.com for more info.
NUMERICA METHODS 1 final touch summary for test 1musadoto
MY FINAL TOUCH SUMMARY FOR TEST 1
ON 6TH MAY 2018
TOPICS AND MATERIALS COVERED
1. Class lecture notes (Basic concepts, errors and roots of function).
2. Lecture’s examples.
3. Past Years Examples.
4. Past Years examination papers.
5. Tutorial Questions.
6. Reference Books + web.
College Algebra MATH 107 Spring, 2016, V4.7 Page 1 of .docxclarebernice
College Algebra MATH 107 Spring, 2016, V4.7
Page 1 of 11
MATH 107 FINAL EXAMINATION
This is an open-book exam. You may refer to your text and other course materials as you work
on the exam, and you may use a calculator. You must complete the exam individually.
Neither collaboration nor consultation with others is allowed.
Record your answers and work on the separate answer sheet provided.
There are 30 problems.
Problems #1–12 are Multiple Choice.
Problems #13–21 are Short Answer. (Work not required to be shown)
Problems #22–30 are Short Answer with work required to be shown.
MULTIPLE CHOICE
1. Determine the domain and range of the piecewise function. 1. ______
A. Domain [–1, 3]; Range [–3, 1]
B. Domain [–1, 1]; Range [–1, 3]
C. Domain [–1/2, 0]; Range [–1, 0]
D. Domain [–3, 1]; Range [–1, 3]
2. Solve: 17 3x x+ = − 2. ______
A. No solution
B. −1
C. −7
D. −1, 8
2 4 -4
-2
-4
2
4
-2
College Algebra MATH 107 Spring, 2016, V4.7
Page 2 of 11
3. Determine the interval(s) on which the function is increasing. 3. ______
A. (–2, 4)
B. (–∞, –2) and (4, ∞)
C. (–3.6, 0) and (6.7, ∞)
D. (–3, 1)
4. Determine whether the graph of 7y x −= is symmetric with respect to the origin,
the x-axis, or the y-axis. 4. ______
A. not symmetric with respect to the x-axis, not symmetric with respect to the y-axis, and
not symmetric with respect to the origin
B. symmetric with respect to the x-axis only
C. symmetric with respect to the y-axis only
D. symmetric with respect to the origin only
5. Solve, and express the answer in interval notation: | 6 – 5x | ≤ 14. 5. ______
A. [–8/5, 4]
B. (–∞, −8/5] ∪ [4, ∞)
C. (–∞, –8/5]
D. [4, ∞)
College Algebra MATH 107 Spring, 2016, V4.7
Page 3 of 11
6. Which of the following represents the graph of 8x + 3y = 24 ? 6. ______
A. B.
C. D.
College Algebra MATH 107 Spring, 2016, V4.7
Page 4 of 11
7. Write a slope-intercept equation for a line parallel to the line x – 7y = 2 which passes through
the point (14, –9). 7. ______
A.
1
7
7
y x= − −
B. 7 89y x= − +
C.
1
9
7
y x= −
D.
1
11
7
y x= −
8. Which of the following best describes the graph? 8. ______
A. It is the graph of a function and it is one-to-one.
B. It is the graph of a function and it is not one-to-one.
C. It is not the graph of a function and it is one-to-one.
D. It is not the graph of a function and it is not one-to-one.
College Algebra MATH 107 Spring, 2016, V4.7
Page 5 of 11
9. Express as a single logarithm: 5 log y – log (x + 1) + log 1 9. ______
A.
log(5 )
log( 1)
y
x +
B. ( )log 5 y x−
C.
5
log
1
y
x
+
D.
5 ...
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
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.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
2. By the end of this lesson you
should know how to solve
• Linear Inequalities
• Quadratic Inequalities
• Rational Inequalities
3. What do you know about inequalities?
What are differences between them and
equations?
Do inequalities have special properties?
The goal of such questions is to:
Understand what students are thinking.
Elicit several answers from the students.
4. Properties of Inequality
Let a, b and c represent real numbers.
1. If a < b, then a ± c < b ± c.
2. If a < b and if c > 0, then ac < bc.
3. If a < b and if c < 0, then ac > bc.
4.If a < b and if c > 0, then a/c < b/c.
5. If a < b and if c < 0, then a/c > b/c.
Replacing < with >, ≤ , or ≥ results in similar properties.
5. Can we classify inequalities?
According to what?
Encourage students to jot down answers, then
combine answers in a small group.
6. Linear Inequality in One
Variable
A linear inequality in one variable is
an inequality that can be written in the
form
where a and b are real numbers, with
a ≠ 0. (Any of the symbols ≥, <, and ≤
may also be used.)
0,ax b+ >
7. As we did with linear equations,
we are looking for the solution set
of the linear inequality.
Invite students to talk about linear eq., other types of
eq’s., and refresh their memories about sets as well
as the concept of sol. set.
8. Example SOLVING A LINEAR INEQUALITY
Solve
Solution
3 5 7.x− + > −
3 5 7x− + > −
53 7 55x− > −− −+ Subtract 5.
3 12x− > −
3 12
3 3
x
<
− −
− −
Divide by –3. Reverse
the direction of the
inequality symbol
when multiplying or
dividing by a negative
number.
Don’t forget to
reverse the
symbol here.
< = −∞4 ( ,4)x
Combine like terms.
Ask them, why is this linear?
9. Here (and after any major point) one can pause and show
one or two multiple-choice questions, call for a vote (turn
this into a competition or a debate).
Solve
A.
B.
C.
D.
2
12 4
5
x
x
−
≥ −
10
3
x ≤
30
11
x ≤
10
3
x ≥
30
11
x ≥
10. Application FINDING THE BREAK-EVEN
POINT
If the revenue and cost of a certain product
are given by
4R x= and 2 1000,C x= +
where x is the number of units produced and
sold, at what production level does R at least
equal C ?
Let them try and guide them to the right thinking.
11. Implementing applications is attracting students and showing the beauty
of mathematics. Let alone its crucial role in satisfying Bloom’s taxonomy
12. Application FINDING THE BREAK-EVEN
POINT
Solution For R at least equal C , this means R ≥ C .
4 2 1000x x≥ + Substitute.
2 1000x ≥ Subtract 2x.
500x ≥ Divide by 2.
The break-even point (all costs that must be paid are paid,
and there is neither profit nor loss) is at x = 500. This
product will at least break even if the number of units
produced and sold is in the interval [500, ∞).
13. When finishing some part of the lecture, allow time for questions.
Doing so helps students to sort out the information. Then ask a
question that motivates the next part.
What if the inequality isn’t linear?
What if we are dealing with quadratic inequality?
What does quadratic inequality look like?
How can we solve it?
This kind of questions motivates students and plays a fundamental
role in brainstorming.
14. Quadratic Inequalities
A quadratic inequality is an inequality
that can be written in the form
2
0ax bx c+ + <
for real numbers a, b, and c, with a ≠
0. (The symbol < can be replaced with
>, ≤, or ≥.)
15. Solving a Quadratic Inequality
Step 1 Solve the corresponding quadratic
equation.
Step 2 Identify the intervals determined by
the solutions of the equation.
Step 3 Use a test value from each interval
to determine which intervals form
the solution set.
16. Example SOLVING A QUADRATIC
INEQULITY
Solve
Solution
2
12 0.x x− − <
Step 1 Find the values of x that satisfy
x2
– x – 12 = 0.
( 3)( 4) 0x x+ − = Factor.
3 0x + = 4 0x − =or Zero-factor property
3x = − 4x =or Solve each equation.
Ask students: How can we solve this equation?
17. Example SOLVING A QUADRATIC
INEQULITY
Step 2
–3 0 4
Interval A Interval B Interval C
(–∞, – 3) (–3, 4) (4, ∞)
18. Example SOLVING A QUADRATIC
INEQULITY
Interval
Test
Value
Is x2
–x – 12 < 0
True or False?
A: (–∞, –3) –4 (–4)2
– (–4) – 12 < 0 ?
8 < 0 False
B: (–3, 4) 0 02
– 0 – 12 < 0 ?
–12 < 0 True
C: (4, ∞) 5 52
– 5 – 12 < 0 ?
8 < 0 False
Step 3 Choose a test value from each interval.
Since the values in Interval B make the inequality
true, the solution set is (–3, 4).
19. Group work SOLVING A QUADRATIC
INEQUALITY
Solve the inequality
Solution
2
2 5 12 0.x x+ − ≥
Step 1 Find the values of x that satisfy
Corresponding
quadratic equation
(2 3)( 4) 0x x− + = Factor.
2 3 0x − = or 4 0x + = Zero-factor property
2
2 5 12 0.x x+ − =
Divide students into small groups and let them solve together. This will indeed develop the students’ ability to
communicate and work in teams. After they finish let students choose one of them to go to the board to show their work.
•
20. Group work SOLVING A QUADRATIC
INEQUALITY
Solve
Solution
2
2 5 12 0.x x+ − ≥
Step 1
2 3 0x − = or 4 0x + =
3
2
x = or 4x = −
21. Group work SOLVING A QUADRATIC
INEQUALITY
Solve
Solution
2
2 5 12 0.x x+ − ≥
Step 2 The values form the intervals on
the number line.
–4
0
3/2
Interval A Interval B Interval C
(–∞, – 4] [– 4, 3/2] [3/2, ∞)
22. Group work SOLVING A QUADRATIC
INEQUALITY
Step3 Choose a test value from each interval.
Interval
Test
Value
Is 2x2
+ 5x – 12 ≥ 0
True or False?
A: (–∞, –4] –5 2(–5)2
+5(–5) – 12 ≥ 0 ?
13 ≥ 0 True
B: [– 4, 3/2] 0 2(0)2
+5(0) – 12 ≥ 0 ?
–12 ≥ 0 False
C: [3/2, ∞) 2 2(2)2
+ 5(2) – 12 ≥ 0 ?
6 ≥ 0 True
The values in Intervals A and C make the inequality true, so
the solution set is the union of the intervals
−∞ − ∪ ∞÷
3
( , 4] , .
2
23. Application FINDING PROJECTILE HEIGHT
If a projectile is launched from ground level
with an initial velocity of 96 ft per sec, its
height s (in feet) t seconds after launching is
given by the following equation.
2
16 96s t t= − +
When will the projectile be greater than 80 ft
above ground level?
Here one should invite students to give some thoughts.
Let them participate.
24. Application FINDING PROJECTILE HEIGHT
2
16 96 80t t− + >
Set s greater than
80.
Solution
2
16 96 80 0t t− + − > Subtract 80.
2
6 5 0t t + <− Divide by – 16.
Reverse the
direction of the
inequality
symbol.
Now solve the
corresponding equation.
26. Application FINDING PROJECTILE HEIGHT
Solution
Interval C
1
0
5
Interval A Interval B
(–∞, 1) (1, 5) (5, ∞)
By testing the intervals, we observe that
values in Interval B, (1, 5) , satisfy the
inequality. The projectile is greater than 80
ft above ground level between 1 and 5 sec
after it is launched.
27. Solving a Rational Inequality
Step 1 Rewrite the inequality, if necessary, so that
0 is on one side and there is a single fraction on the
other side.
Step 2 Determine the values that will cause either
the numerator or the denominator of the rational
expression to equal 0. These values determine the
intervals of the number line to consider.
28. Solving a Rational Inequality
Step 3 Use a test value from each interval to
determine which intervals form the solution set.
A value causing the denominator to equal zero will
never be included in the solution set. If the
inequality is strict, any value causing the numerator
to equal zero will be excluded. If the inequality is
nonstrict, any such value will be included.
30. Try
individually
Solve
Solution
5
1.
4x
≥
+
Step 1 5
1 0
4x
− ≥
+
Subtract 1 so that 0 is
on one side.
( )5 4
0
4
x
x
− +
≥
+
Write as a single fraction.
SOLVING A RATIONAL INEQUALITY
Solving individually improves the ability to concentrate
easier and work faster and helps in getting the whole credit
for the work. (Some students prefer this)
1
4
0
x
x
−
+
≥
Combine terms in the
numerator, being careful
with signs.
31. Example
Solve
Solution
5
1.
4x
≥
+
Step 2 The quotient possibly changes sign
only where x-values make the numerator or
denominator 0. This occurs at
1 0x− = or 4 0x + =
1x = or 4x = −
SOLVING A RATIONAL INEQUALITY
Try individually
33. HOMEWORK 4
Step 3 Choose test values.
Interval
Test
Value
A: (–∞, –4) – 5
B: (–4, 1] 0
C: [1, ∞) 2
5
Is 1 True or False?
4x
≥
+
?
5
1
5 4
≥
− +
5 1 False− ≥
0
?
5
1
4
≥
+ 5
1 e
4
Tru≥
2
?
5
1
4
≥
+
5
1 e
6
Fals≥
SOLVING A RATIONAL
INEQUALITY
Try individually
34. Example 8
Step 3
The values in the interval (–4, 1) satisfy the
original inequality. The value 1 makes the
nonstrict inequality true, so it must be included
in the solution set. Since –4 makes the
denominator 0, it must be excluded. The
solution set is (–4, 1].
SOLVING A RATIONAL INEQUALITYTry
individually
35. Today’s lesson has finished
Your leaving ticket is to
sum up
Ask students to say one or two sentences about the lesson before
they leave.