SLC- S22/W2: Equations and Inequalities

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Hello friends and welcome to my article in this week's edition of SLC S22/W2 I appreciate @khursheedanwar I would be putting in efforts to bring forth valid answers to the questions in the task.


"Explain difference between linear and quadratic equations. Provide examples of each type of equations and describe their general forms."

Linear Equations:

These are special type of equations bearing high degree present in the main variable. They could be represented in a graph portraying a straight line graph in a graphic representation, joining each point of opposite direction say x and y for horizontal and vertical axes respectively. A good example of linear equations would be ax+ y = 0

Quadratic Equations:

This is an equation in which the superscript of any coefficient is 2. They are basically having three expressions equated to 0 which makes them an equation. A quadratic equations form would be like ax²+bx+C=0 where x, x² remains constant coefficient and a,b,C all are variables which could be changed at any point in time, dependent on the question tasked.

Real life scenario;

Linear equations: A business driver charges $10 over 20km and an additional $4 on each double distance covered. Write linear equations representing total fare for 50km ride.

Quadratic Equations: A pendulum bob is swung in quick succession, twice it's speed and thus a new record revealed, we could assume a quadratic equation from this and it would be x² +2x + 4= 0


"Describe two different types of inequalities ( which are not explained in course) provide examples of each type of inequality and explain how to solve them"

Quadratic Inequalities:
Inequalities simply implies the quadratic expression does not equate or is not equal to, therefore they are quadratic expressions due to the fact that they do not equate to 0. The are either <, ≤, >, ≥ These are special signs to look out for in quadratic inequalities, incorporating into the quadratic equations. A quadratic expression would be in the form of...
ax² + bx + C < 0
ax² + bx + C ≤ 0
ax² + bx + C > 0
ax² + bx + C ≥ 0

A good example would be
x² + x - 8 < 0
Sorting out factors of -8 we should have (1,-8) (2, -4) (1, 8) (2, 4) Therefore the best factors would definitely be;

(2, -4)
Substituting them into our equation we have;

x² +2x -4x -8 < 0

Factorizing
x(x+2) -4(x+2) < 0
(x+2) (x-4) < 0
Thus; x < 2, x < 4

Absolute inequalities:
These are inequalities with magnitude signs assigned to them. It would be best explained with examples which would be shown below

(2x²-10)² < (12)²
Simplifying;

(2x-10)(2x-10) < (12)²
4x²-20x-20x-100 < 144
4x²-40x-100 < 144
4x²- 40x < 144 +100
4x² - 40x < 244
4x²- 40x -244 < 0

This is an example of an absolute value inequality.


"Solve the following linear equation: 2x+5= 11 show step by step solution and share practical examples of how this equation can be applied in real life scenario"

Solve 2x + 5 = 11
We pick out like terms

2x = 11-5
`2x = 6'
Dividing both sides by the coefficient of x which is 2 we have x= 3

real life application
A total number of 11 students in a class about 5 of them are girls and x are boys the number of boys remains anonymous. Extra 2 females were put in the same row as some boys a week later, how many boys are in the class?

Solution:
Total number of students = 11
number of boys = x
Girls observed = 2 (x) and 5 in that row

Mathematically;
2x +5 = 11
2x = 11-5
2x = 6
Diving by coefficient of x
x = 3
Therefore, there are about 3 boys in the class which sat in the row with the girls


• Task 4
Scenario: Tom's Bakery

Tom's bakery is a most famous bakery of tow and it's famous for its yummiest pastries and bread. Tom which is owner want for making sure that he has sufficient ingredients for meeting needs for his famous chocolate cake and chocolate cake recipe needs 2 cups of flour, 1 cup of sugar and 1/2 cup of cocoa powder for each cake.


Task 4word problem
EquationIf Tom wants for making x cakes and he has 10 cups of flour, 8 cups of sugar and 4 cups of cocoa powder then calculate number of cakes that can be made by Tom? Inequality Tom wants for making sure that he has sufficient sugar for meeting needs for his chocolate cake. He also knows that each cake needs 1 cup of sugar and he has 8 cups of sugar which are available. He also knows that he wants to make at least 6 cakes. Solve for equations and inequality both from above scenario

Case 1: Tom Bakery

Requirements: 2 cups of flour, 1 cup of sugar, ½ cup of cocoa powder.

Availability: 10 cups of flour, 8 cups of sugar, 4 cups of cocoa powder.

To ascertain maximum number of cakes we develop a equation.

Flour : 2x ≤ 10
Sugar : x ≤ 8
Cocoa : ½x ≤ 4

Flour= 2x≤10
x ≤ 10÷2
x ≤ 5

Sugar x ≤ 8

Cocoa ½x ≤ 4

x ≤ 5, 8, 8

Tom would work with minimum of 5 cakes. It turns inequalities when Tom decides to make more than 5 cakes, he would need more sugar to meet it. He would have used up his flour therefore would not be able to make cakes without flour.

Inequality x≤8 ( sugar) x≥6 minimum requirement.

6 ≤x≤8. Tom can only make cake between 6 and 8 if other resources are available


Case 2
Scenerio: Number of gallons Ashley needs.

Suppose there's a person named Ashley who is planning for a road trip from City A to City B.If I talk about distance between two cities then it's around 240 miles. Ashley's car fuel tank have ability of holding 12 gallons of Suppose that his car gets 20 miles for each how many gallons of gas may Ashley purchase if he already has 2 gallons in his tank and he wants to have minimum 1 gallon left over when he arrives at City B?

Equation Let's consider x as number of gallons of gas Ashley can purchase.
Inequality x + 2 ≥ (240/20) - 1


Distance = 240 mils from A-B, 20 miles gallon, 2 gallons ( initial gas tank)
1 gallon ( minimum on arrival)

Total gallon= Total distance / Total gallon per gallon

TG = 240/20
TG = 12

2 Gallons is in possession of Ashly and needs one gallon when he gets to city B, we deribe equation.

x + 2≥ 12- 1
Simplifying
x ≥ 12-1-2
x ≥ 9

Therefore Ashley needs 9 gallons to arrive city B in her tank

Cc; @khursheedanwar

I have concluded my article and invite @enamul17 @nancy0 @mariami @solaymann to join challenge

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