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Topic 2 Equations and Inequalities

In this topic, you will understand and solve problems using equations and inequalities.

Formula

A formula is a concise way of expressing information symbolically, often used to represent a mathematical relationship or equation. For example, the formula for the area of a rectangle is length multiplied by width (A = l * w).

The subject of a formula

The “subject of a formula” refers to the variable or expression that is isolated or solved in a mathematical formula. In other words, it is the variable that you want to express in terms of the other variables in the formula. This process is often called “making a variable the subject of the formula” or “solving for a variable.”

For example, consider the formula for the area of a rectangle:

A=l×w

In this formula, A represents the area, l represents the length, and w represents the width of the rectangle. If you want to make l the subject of the formula, you would rearrange the formula to express l in terms of A and w:

l=A/w​

Now, l is the subject of the formula.

Variable

Variables can be classified into two main types:

  1. Independent Variables: These are variables that you can choose or manipulate in an experiment or mathematical expression. They are often denoted by letters like x or t.
  2. Dependent Variables: These are variables that depend on the independent variables. Their values are determined by the values of the independent variables. In mathematical expressions or formulas, the dependent variable is often denoted by a different letter, such as y or f(x).

For example, in the equation, y=2x+3 x is the independent variable, and y is the dependent variable.

Critical number

A critical number refers to a value in the domain of a function where the derivative is either zero or undefined. Critical numbers are important in the study of functions because they often indicate points where the function may have local extrema (maxima or minima) or points of inflection.

By the end of this topic, you should be able to:

Build formulae from word statements.

Re-write a given formula by changing the subject.

Solve equations and inequalities, representing the solutions on a number line or graphically.

Introduction

In daily life, equations and inequalities are used quite often, for example: when comparing things in terms of quantities, sizes, and so on; and when calculating the amount of money to go out with for a tour.

You may have also come across statements such Achwa” and many others. Have you ever wondered how people come up with such as; “a bird in the hand is worth two in the bush”, “River Nile is longer than River statements?”

In this topic, you will understand and solve problems using equations and inequalities.

Activity 2.0(a) (Work in groups)

(a) Copy and fill all the empty squares in the SUDOKU game. Make sure that the numbers 1 to 9 appear appear exactly once in each row and each column.

(b) Explain how you identified the missing digits.

(c) Compare your results with other members in different groups.

Activity 2.0(b) (Work in groups)

In a certain school, the Senior Four class started lessons at 6:00 a.m., while the Senior Three class started lessons at 8:00 a.m. The Senior Four class has been learning for k hours now

Write down an expression to represent the number of hours for which the Senior Three class has been learning.

2.1 Building Formulae from Word Statements

In solving Mathematical problems, we often use formulae which are already built. In this subtopic, you will explore how to build formulae from word statements.

Activity 2.1(a) (Work in groups)

A teacher awards two marks for each correct answer, plus three marks as an incentive for test attendance. If p is the number of questions attempted and m is the total mark that a learner gained from the test: 14

(a) show how the two variables are related.

(b) tell your class, the minimum and maximum mark for a test which had only nine questions.

Activity 2.1(b) (Work in groups)

Omondi went shopping and bought 4 kg of sugar, 5 bars of soap, and 8 litres of cooking oil. In the evening, on the same day, Nakamatte went to the same supermarket and bought 2 kg of sugar, 8 bars of soap, and 7 litres of cooking oil. Let x, y, and z represent the costs of 1 kg of sugar, 1 bar of soap, and 2 litres of cooking oil, respectively.

(a) Write down the formulae for the total amount of money that each one of them spent on all the items that he/she bought.

(b) If y < x and z <y, who of them spent more money than the other? Justify your answer.

Example 2.1

There are three tall buildings, A, B, and C, on a certain street of Kampala. The average heights of buildings A and B, A and C, and B and C are 60 m, 50 m, and 55 m, respectively.

(a) Write down three formulae from the information above.

(b) By solving, which of the buildings is the tallest?

Solution:

Let the height of building A be u.

Let the height of building B be v.

Let the height of building C be w.

Using any method of solving simultaneous equations that you learnt in Senior Three, the solutions to the system of equations above are obtained as u = 55, v = 65, and w = 45, hence, the tallest building is B.

Exercise 2.1

1. In a car bond, there are minibuses and saloon cars. Each minibus occupies y m2 of space and each saloon car occupies x m2 of space.

(a) Devise a formula for the area occupied by all the vehicles at the car bond.

(b) If the total area of the bond is 20,000 m2, write the equation that relates the total area to your answer in (a) above.

2) Tom plans to buy fencing poles and barbed wire to fence his plot. Each roll of barbed wire costs UGX 90,000 and each pole costs UGX 7,000.

(a) Devise a formula of how much Tom will spend on the fencing exercise.

(b) If Tom spends UGX 400,000 altogether, write the equation that relates the total amount of money he spends to your answer in (a) above.

3) In a market, a certain trader sells a basin of Irish potatoes at UGX 7,000 and a basin of charcoal at UGX 12,000.

(a) Derive a formula for the trader’s sales.

(b) If the trader bought Irish potatoes and charcoal at UGX 180,000 altogether, what profit did he make?

4) A furniture company sells a sofa set at UGX m each and a dining table at UGX n each. On a certain day, the company sold x sofa sets and y dining tables. How much did the company earn from these sales?

5) The figure below shows a cuboid ABCDEFGH. Find its total surface area in terms of I, w and h.

6) A trip of hardcore material for building costs UGX 50,000 and a trip of sand costs UGX 30,000. Waiswa, who wants to build his residential house, goes to buy hardcore and sand. Derive a formula for the total amount of money he spends.

7) Muhangi has a plot of land measuring 30 m by 30 m. He plans to build a house on this plot and the remaining part to be designed into a first-class compound. Write down an equation that connects the areas that will be occupied by the house and the compound.

2.2 Re-writing a Given Formula by Changing the Subject

Formulae are usually written in such a way that one variable or quantity is expressed in terms of the others.

Activity 2.2(a) (Work in groups)

(a) Write down the formula for the volume of a cuboid.

(b) Re-arrange the formula written in (a) above to form three (3) more equations.

Follow the examples below:

2.3 Solving Equations and Inequalities, Representing the Solutions on a Number Line or Graphically

Solving an equation or inequality means to find the value or values of the unknown or unknowns that satisfies or satisfy the equation or inequality at hand.

Solving Equations

The equations can be either linear (in one or two unknowns) or quadratic (in one unknown).

Activity 2.3(a) (Work in groups)

Lumu is a poultry farmer and he rears chickens for sale. One Sunday morning, he took 64 chickens to the market. He sold the chickens at UGX 13,000 each and he only sold out less than half of them. In the afternoon, he reduced the price of each chicken, but the price was still a multiple of 1,000. He sold all the chicken. He UGX 712,000 from both the morning and afternoon sales. Assess the number of chicken that Lumu sold in the morning. got

Quiz: The equation s – 3 = s + 3 has no solution. Is this statement correct? Explain your answer.

Exercise 2.3

  1. John is five years older than Peter, James is twice as old as John, and the sum of all their ages is 55. What are their individual ages?
  2. In the Beginning of Term examinations, Joan got twice as many marks as Phiona and ten marks more than Juma. The total of their marks was 170. What was Phiona’s mark?
  3. Mpagi is five years younger than Kawere. In ten years’ time, the sum of their will be 49. What are their current ages? ages
  4. The sum of three consecutive counting numbers is 48. What are the numbers
  5. The mean of 2, 3, 4, (x + 1), and 6 is 7.5. What is the value of x?
  6. Think of a number, subtract 5 from it, and divide the result by 2. The answer is 10. What is the number?
  7. A Primary Four learner scored 56, 64, and 73 in three papers. What should be his / her score in the fourth paper, so that the average score of all the four papers is 66?
  8. The following marks arranged in order of their increasing magnitude, were obtained in a Mathematics test. 15, 16, 18, x + 3, 25, 26. If their median is 20, find the value of x.
  9. Two consecutive integers are such that the difference of their squares is 89. What are these integers?
  10. Solve the following equations.

Exercise 2.4

  1. Find the greatest possible values of two whole numbers, such that one of them is five times the other and that their sum is at most 54.
  2. The sum of three consecutive counting numbers is at least 45. Find the least possible values of these numbers.
  3. Derrick’s height is 20 cm more than John’s. The sum of their heights is at least 360 cm. Find the least possible heights that Derrick and John can have.
  4. Two whole numbers are such that one of them is greater than the other by five. Their integral quotient is at least 2. Find the greatest possible values of these numbers.
  5. The length of a rectangle is 4 cm more than its width. Find the greatest possible length and width if the perimeter of the rectangle does not exceed 68 cm.
  6. Two pieces of wood are such that the length of one of them is 10 cm more than that of the other. The total length of the pieces is at most 100 cm. Find the greatest possible length of each piece.
  7. The Fahrenheit and Celsius temperature scales are related by the formula, F = 20 C + 32. What range of C corresponds to 98° < F≤ 144°?
  8. The area occupied by a bus in a park is 5 m2 more than that occupied by a taxi. The total area occupied by both the bus and the taxi does not exceed 45 m2. Find the greatest possible area that each of the taxi and the bus occupies.

Project Work:

Identify one scenario and describe how inequalities are applicable in real-life situations. You can attach a photograph or a video to give support to your work. Present your work to the rest of the class. ?

6) The length and width of a rectangle are (2x + 3) cm and (x-4) cm, respectively. The perimeter of the rectangle is 40 cm. Find the value of x.

7) Joan is three times as old as Josephine. In five years’ time, Joan’s age will be twice Josephine’s, then. What are their current ages?

8) The difference between two whole numbers is 4 and the difference between their squares is at least 144. Find the least possible values of these whole numbers.

Topic Summary

In this topic, you have learnt:
1) how to build formulae from word statements and how to change the subject of a formula.
2) that a subject of a formula is the single variable (usually on the left of the equal signs) that everything else in that formula is equal to.
3) how to solve linear inequalities. You found out that a linear equation has almost one solution.
4) that a linear inequality is similar to a linear equation and has the power of the variable equal to 1.
5) that when solving inequalities, dividing or multiplying on both sides of an
6) inequality by a negative number, reverses the inequality sign.
7) that if the inequality symbol is S or 2, use a solid line when drawing the graph that if the inequality symbol is < or >, use a dotted line when drawing the
graph.

Assignment

Sample Activity of Equations and Inequalities

ASSIGNMENT : Sample Activity of Equations and Inequalities MARKS : 10  DURATION : 1 week, 3 days

 

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