Linear equations in two variables. Please download the powerpoint file to enable animation.
Disclaimer: Some parts of the presentation are obtained from various sources. Credit to the rightful owners.
Linear equations in two variables. Please download the powerpoint file to enable animation.
Disclaimer: Some parts of the presentation are obtained from various sources. Credit to the rightful owners.
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We can define heteroscedasticity as the condition in which the variance of the error term or the residual term in a regression model varies. As you can see in the above diagram, in the case of homoscedasticity, the data points are equally scattered while in the case of heteroscedasticity, the data points are not equally scattered.
Two Conditions:
1] Known Variance
2] Unknown Variance
We can define heteroscedasticity as the condition in which the variance of the error term or the residual term in a regression model varies. As you can see in the above diagram, in the case of homoscedasticity, the data points are equally scattered while in the case of heteroscedasticity, the data points are not equally scattered.
Two Conditions:
1] Known Variance
2] Unknown Variance
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We can define heteroscedasticity as the condition in which the variance of the error term or the residual term in a regression model varies. As you can see in the above diagram, in the case of homoscedasticity, the data points are equally scattered while in the case of heteroscedasticity, the data points are not equally scattered.
Two Conditions:
1] Known Variance
2] Unknown Variance
We can define heteroscedasticity as the condition in which the variance of the error term or the residual term in a regression model varies. As you can see in the above diagram, in the case of homoscedasticity, the data points are equally scattered while in the case of heteroscedasticity, the data points are not equally scattered.
Two Conditions:
1] Known Variance
2] Unknown Variance
This session provides a comprehensive overview of the latest updates to the Uniform Administrative Requirements, Cost Principles, and Audit Requirements for Federal Awards (commonly known as the Uniform Guidance) outlined in the 2 CFR 200.
With a focus on the 2024 revisions issued by the Office of Management and Budget (OMB), participants will gain insight into the key changes affecting federal grant recipients. The session will delve into critical regulatory updates, providing attendees with the knowledge and tools necessary to navigate and comply with the evolving landscape of federal grant management.
Learning Objectives:
- Understand the rationale behind the 2024 updates to the Uniform Guidance outlined in 2 CFR 200, and their implications for federal grant recipients.
- Identify the key changes and revisions introduced by the Office of Management and Budget (OMB) in the 2024 edition of 2 CFR 200.
- Gain proficiency in applying the updated regulations to ensure compliance with federal grant requirements and avoid potential audit findings.
- Develop strategies for effectively implementing the new guidelines within the grant management processes of their respective organizations, fostering efficiency and accountability in federal grant administration.
Understanding the Challenges of Street ChildrenSERUDS INDIA
By raising awareness, providing support, advocating for change, and offering assistance to children in need, individuals can play a crucial role in improving the lives of street children and helping them realize their full potential
Donate Us
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Presentation by Jared Jageler, David Adler, Noelia Duchovny, and Evan Herrnstadt, analysts in CBO’s Microeconomic Studies and Health Analysis Divisions, at the Association of Environmental and Resource Economists Summer Conference.
Many ways to support street children.pptxSERUDS INDIA
By raising awareness, providing support, advocating for change, and offering assistance to children in need, individuals can play a crucial role in improving the lives of street children and helping them realize their full potential
Donate Us
https://serudsindia.org/how-individuals-can-support-street-children-in-india/
#donatefororphan, #donateforhomelesschildren, #childeducation, #ngochildeducation, #donateforeducation, #donationforchildeducation, #sponsorforpoorchild, #sponsororphanage #sponsororphanchild, #donation, #education, #charity, #educationforchild, #seruds, #kurnool, #joyhome
ZGB - The Role of Generative AI in Government transformation.pdfSaeed Al Dhaheri
This keynote was presented during the the 7th edition of the UAE Hackathon 2024. It highlights the role of AI and Generative AI in addressing government transformation to achieve zero government bureaucracy
A process server is a authorized person for delivering legal documents, such as summons, complaints, subpoenas, and other court papers, to peoples involved in legal proceedings.
2. LESSON
OBJECTIVES
•Defines the terms
‘linear inequality in
one variable’.
•Differentiates linear
inequality in one
variable from linear
equation in one
variable.
KNOWLEDGE
•Illustrates a
linear inequality
in one variable.
SKILLS
• Appreciates the
representation of
linear inequalities in
one variable in real
life.
•Cooperates with
other group
members during
group activities.
ATTITUDE
4. 𝟏. 𝒙 + 𝟒 = 𝟖
𝟐. 𝟏𝟖𝒙 − 𝟏𝟐 > 𝟐
𝟑. 𝟕𝒙 𝟏𝟒 − 𝟑
𝟒. 𝟐𝒙 ≤ 𝟗
𝟓. 𝒙 +
𝟏
𝟐
≥ 8 linear inequality in one variable
linear equation in one variable
linear inequality in one variable
algebraic expression
linear inequality in one variable
5. 𝟏. 𝒙 + 𝟒 = 𝟖
𝟐. 𝟏𝟖𝒙 − 𝟏𝟐 > 𝟐
𝟑. 𝟕𝒙 𝟏𝟒 − 𝟑
𝟒. 𝟐𝒙 ≤ 𝟗
𝟓. 𝒙 +
𝟏
𝟐
≥ 8
What is the symbol used in item 1 for it
to be classified as a linear equation?
How did you know that the expression
in item 3 is an algebraic expression?
What is the symbol used in items 2, 4
and 5 for them to be classified as linear
inequalities?
Aside from the symbols used in items
2, 4, and 5, what is the other symbol for
inequality?
6. Click to change subtitle
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Edit title
On the other hand, if an expression relates two
expressions or values with a ‘<’ (less than) sign, ‘>’
(greater than) sign, ‘≤’ (less than or equal) sign or ‘≥’
(greater than or equal) sign, then it is called as an
Inequality
Mathematical/ Algebraic expressions help us convert
problem statements into entities and thus, help solve them. If
the expression equates two expressions or values, then it is
called an equation.
For e.g. 3x + 5y = 8. If it is a linear equation in one
variable, an example is 6x-7=0.
6
7. “
7
Mind the symbols
A linear inequality in
one variable is an inequality
which can be put into the form
𝒂𝑥 + 𝑏 > 𝑐, where a, b, and c
are real numbers.
Note that the “> "can be
replaced by ≥, <, or ≤.
8. “
8
Examples:
2𝑥 > 4
2𝑥 − 2 < 6𝑥 − 5 𝑐𝑎𝑛 𝑏𝑒 𝑤𝑟𝑖𝑡𝑡𝑒𝑛 − 4𝑥 + −2 < −5.
Then, it would be -4x - 2< -5.
6𝑥 + 1 ≥ 3 𝑥 − 5 𝑐𝑎𝑛 𝑏𝑒 𝑤𝑟𝑖𝑡𝑡𝑒𝑛 𝑎𝑠 6𝑥 + 1 ≥ −15
How would each look like if it were
in standard form?
9. “
9
How would each look like if it were in
standard form?
2𝑥 > 4
-4x - 2< -5
6𝑥 + 1 ≥ −15
𝐴𝑛𝑠𝑤𝑒𝑟: 2𝑥 − 4 > 0
𝐴𝑛𝑠𝑤𝑒𝑟: 4𝑥 + 3 > 0
𝐴𝑛𝑠𝑤𝑒𝑟: 6𝑥 + 16 > 0
10. “
10
Activity 1: Clap twice if the equation or inequality in one variable
is in standard form and clap once if it’s not.
𝑥 ≥ 5
2-5x=6
2(3x-4) >8
3x-15=
2x+2≤ 0
11. “
STEPS IN TRASFORMING ANY LINEAR INEQUALITY IN ONE
VARIABLE INTO STANDARD FORM
Let us rewrite the following linear
inequalities in the form ax+ b>0.
𝑎. 2𝑥 − 1 < 3𝑥 + 5
𝑏. −
𝑥
3
+ 3 ≥ 4
12. “
STEPS IN TRASFORMING ANY LINEAR INEQUALITY IN ONE
VARIABLE INTO STANDARD FORM
Step 1: First, we need to remove the
fraction/s by multiplying each term by
the LCM (Least Common Multiple )if there
is/ are fraction/s in the given inequality.
𝑎. 2𝑥 − 1 < 3𝑥 + 5
Since there is no fraction present in this
inequality, we will move on to step 2.
13. “
STEPS IN TRASFORMING ANY LINEAR INEQUALITY IN ONE
VARIABLE INTO STANDARD FORM
Step 2: Next, apply the necessary properties of
inequality so that only zero is left on the right side
of the inequality.
2𝑥 − 1 < 3𝑥 + 5
-3x
-x - 1 < 5
− 5 − 5
−𝑥 − 6 < 0
14. “
STEPS IN TRASFORMING ANY LINEAR INEQUALITY IN ONE
VARIABLE INTO STANDARD FORM
Step 3: If the leading coefficient, a is negative,
then, multiply Then, since our a term is negative,
we multiply everything by −1 to make it positive.
(−1)(−𝑥 − 6) < 0(−1)
𝑥 + 6 < 0
Where:
a=1, b=6
15. “
STEPS IN TRASFORMING ANY LINEAR INEQUALITY IN ONE
VARIABLE INTO STANDARD FORM
First, we need to remove the fraction/s by
multiplying each term by the LCM.
b. −
𝑥
3
+ 3 ≥ 4
(3)(−
𝑥
3
+ 3) ≥ (4)(3)
−𝑥 + 9 ≥ 12
b. −
𝑥
3
+ 3 ≥ 4
16. “
STEPS IN TRASFORMING ANY LINEAR INEAQUALITY IN ONE
VARIABLE INTO STANDARD FORM
−𝑥 + 9 ≥ 12
Since 9 and 12 are ‘like’ terms, we can combine
the two integers so we’re left with zero on the right
side of the equation. We have:
−𝑥 + 9 ≥ 12
- 12 ≥ −12
−𝑥 − 3 ≥ 0
17. “
STEPS IN TRASFORMING ANY LINEAR INEAQUALITY IN ONE
VARIABLE INTO STANDARD FORM
Then, since our a term is negative, we
multiply everything by −1 to make it
positive.
(−1)(−𝑥 − 3) ≥ (0)(−1)
𝑥 + 3 ≥ 0
18. “
STEPS IN TRASFORMING ANY LINEAR INEAQUALITY IN ONE
VARIABLE INTO STANDARD FORM
Since we multiplied the answer by -1,
the symbol would be changed into ‘≤’.
Hence, we have:
𝑥 + 3 ≤ 0
Based on the equation,
a=1, b=3
19. “
STEPS IN TRASFORMING ANY LINEAR INEAQUALITY IN ONE
VARIABLE INTO STANDARD FORM
Let us rewrite the following linear
inequalities in the form ax+ b>0 and
determine the a and b values.
2𝑥 − 1 < 3𝑥 + 5
Answer:
𝑥 + 6 > 0 a=1, b= 6
20. 1. What is a linear inequality in one variable?
Questions:
2. How can we identify if the expression is written in
standard form?
3. How can we transform −
1
2
𝑥 − 7 ≥ 5 into standard form?
4. If a=6 and b=7, what would be the standard for of the
linear equation in one variable? How about the linear
inequality in on variable if the inequality symbol is ≥ ?
21. Group Activity: Determine whether each of the following
situations is a linear inequality or a linear equation in
one variable and write mathematical model that
represents the equation or inequality.
Real-life
Situation
Classification
(Linear Inequality
in one variable or
Linear inequality in
one variable)
Mathematical
model
1. Non-metals
(n)have more than 4
valence electrons.
Linear
inequality in
one variable
𝑛 > 4
22. Real-life Situation
Classification (Linear
Inequality in one
variable or Linear
inequality in one
variable)
Mathematical
model
2. For water to
stay liquid, it
should be less
than 32F in
temperature (t).
Linear
inequality
in one
variable
𝑡 < 32
23. Real-life Situation
Classification (Linear
Inequality in one
variable or Linear
inequality in one
variable)
Mathematical
model
3. The maximum
time (t) of
grilling ½ kilo of
meat is 8
minutes.
Linear
inequality
in one
variable
𝑡 ≤ 8
24. Real-life Situation
Classification (Linear
Inequality in one
variable or Linear
inequality in one
variable)
Mathematical
model
4. According to
dieticians, adults
should have no more
than 2.5 tablespoons
(t) of any sugar
added to a food or
drink in a day.
Linear
inequality
in one
variable
𝑡 ≤ 2.5
25. Real-life Situation
Classification (Linear
Inequality in one
variable or Linear
inequality in one
variable)
Mathematical
model
5. The population
(P) of the
Philippines is
about 103, 000,000
P= 103,000,000
Linear
equation in
one variable
26. Things to Remember
If an expression relates two
expressions or values with a ‘<’
(less than) sign, ‘>’ (greater than)
sign, ‘≤’ (less than or equal) sign
or ‘≥’ (greater than or equal) sign,
then it is called as an Inequality.
27. Things to Remember
The steps in rewriting an inequality in one
variable into standard form are as follows:
1. Remove the fraction/s by multiplying each term
by the LCM.
2. Combine ‘like terms’ so what’s left is zero on the
right side of the equation.
3. The a term or the leading coefficient must be
positive. If it is negative, we multiply everything
by −1 to make it positive.
