Absolute Values

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This chapter is all about absolute values. Absolute value is a number regardless of its sign. In other words, absolute value is only how far or how near a number is to zero.

Examples:

1. |3|= 3


2. |9| = 9


3. |-5| = 5


4. |-34| = 34


5. |0| = 0


6. - |45| = - 45


7. |-786| = 786


8. |67| = 67


9. |-1| = 1

How to do math and solve absolute value

It is easier to solve and do math when the given number is an absolute value. Think about it, you don’t have to think of the negative or positive sign of the number. Below are some of the examples in solving absolute numbers.

Addition

There are no complicated steps in mind. Just add both numbers!

Examples:

1. |4| + |5| = |9|
   = 9
2. |67| + |3| = |70| = 70
3. |45| + |8| = |53| = 53
4. |7| + |23| = |30| = 30
5. |90| + |10| = |100| = 100

Subtraction

In subtracting between absolute values, the negative and positive signs are disregarded. We only find the difference between the absolute numbers.

Example:

1. |3-8| = |5| = 5
2. |56 – 2| = |54| = 54
3. |203 – 300| = |97| = 97
4. |45| - |5| = |40| = 40
5. |78| - |23| = |55| = 55

Multiplication

In this process, the signs are also disregarded. We just multiply the numbers.

Example:

1. |8 x -9| = |72| = 72
   
2. |3 x 3| = |9| = 9
3. |25 x 3| = |75| = 75
4. |6 x 3| = |18| = 18
5. |-5 x 2| = |10| = 10
   
   
   

Evaluate the following expressions.

1. 3 |4n| + 8
   When  n= -3
   

   Solution:

   = 3 |4 (-3)| + 8
   = 3 |-12| + 8
   = 3 (12) + 8
   = 36 + 8
   = 44
   
.

2. 6 |x| * 3 |y| + 2 |x+y*z|
   When  x = -2, y = 5, z = 3
   

   Solution:

   = 6 |-2| * 3 |5| + 2 |-2+5*3|
   = 6 (s) * 3 (5) + 2 (13)
   = 12 * 15 + 26
   = 206
   
   
   

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Graph of a Linear Function

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The linear function can be written as f(x) = mx + b in function form or y = mx + b in equation form, where the parameters m and b are real constants and x is a real variable. The constant m is often called the slope of the line or gradient, while b gives the value of the y-intercept, which gives the point of intersection between the graph of the function and the y-axis. Changing these parameters affects their graphs. For example the equation y = x + 2, where m = 1 and b = 2 has the graph

Suppose we change the value of m into m = 2. Our equation becomes y = 2x + 2 and has the graph

As you can see, the line y = 2x +2 is steeper than the line y = x + 2. Now, suppose we change the value of b from the first equation into b = 5, we have the equation y = x + 5. The graph of this linear equation is

From this graph, we can say that the line y = x + 5 translates the line y = x + 2 by 3 steps upward. Hence, changing m makes the line steeper or shallower, while changing b moves the line up or down. We now know that in the case of a linear function y = f(x), expressed in slope-intercept form y = mx + b, m and b are parameters. Also, f(x) = kx represents a direct variation (proportional relationship).

Functions Involving Absolute Value

From the definition of the absolute value of a number, the equation y = |x| is given by

Sketching the graph of the equation y = |x|, we have

Notice that y = |x| is a combination of two linear equations:

(1) y = x, where m = 1 and b = 0, and (2) y = -x, where m = -1 and b = 0. Since these equations are both linear, we follow the same rule as what we have learned about changing the parameters m and b of a linear equation. Changing m would make the lines steeper or shallower, while changing b would translate the lines upward or downward. For example,

The graph of this equation is

Try these questions

1. Which equation does this graph represent?             a. y = |x+3|+3 b. y = |x|+3 c. y = |x+3| d. y = |x - 3| - 3         2. What is the difference between the graph of y=|x + 3| +3 and the graph of y =|x|? Recall that the graph of y = |x| is             a. The graph of y = |x| is steeper than that of y = |x+3|+3. b. The graph of y = |x+3|+3 is same as the graph of y = |x|. c. The graph of y=|x| is translates the graph of y = |x+3|+3 by 3 units to the left and 3 units upward. d. The graph of y=|x+3|+3 translates the graph of y =|x| by 3 units to the left and 3 units upward.         3. Complete the statement: In a linear equation y = mx + b,       a. the parameter b is equal to the y-intercept b. the parameter m is equal to the slope of the line c. when the parameter m increases, the slope of the line becomes steeper d. All of the above.     ANSWERS TO PRACTICE TEST QUESTIONS   A.            y = |x+3|+3. D.            The graph of y=|x+3|+3 translates the graph of y =|x| by 3 units to the left and 3 units upward. D.            All of the above. All statements are true about the linear equation as discussed in the lesson.

Abacus

Abacus

 

Abacus, also called a counting frame, is the first calculating device invented by man for doing basic arithmetic calculations. It was in use centuries before the adoption of the written modern numeral system and is still widely used by merchants, traders and clerks in Asia, Africa, and elsewhere.

Today, abaci are often constructed as a bamboo frame with beads sliding on wires, but originally they were beans or stones moved in grooves in sand or on tablets of wood, stone, or metal. The user of an abacus is called an abacist.

In our Abacus Training Program, only basic arithmetic’s (Addition, Subtraction, Multiplication and Division) is taught. By using basic arithmetic’s, we aim to develop the Concentration, Listening Skills, Photography Memory, Visualization, Mental calculation abilities, Speed & Accuracy. All these skills helps in boosting the child’s Self confidence, which will lead to Overall development of the child.

Mandarin

Mandarin 

 

Our Mandarin courses are professional and innovative.  We believe in making language learning interactive, diverse and fun as well as informative.

In fact, our bank of native Chinese tutors are qualified, skilled and practiced at giving our students a great online experience.

We offer a beginner to advanced course and incorporate the following areas:

Basic Chinese

Our basic Chinese course introduces you to the principles of ‘pinyin’ (a widely-used method of writing Mandarin Chinese using the Latin alphabet). We also help you to master the four tones and understand Chinese pronouns, numbers, greetings, telling the time, the Mandarin calendar, saying goodbye, colours, directions, nationalities and geography. The course is intended as a ‘taster’ for the rudimentary essentials of this language.  

Business Chinese

Our Business Chinese course covers different subject topics as you progress.

Level 1: Will give those with limited knowledge a foundation in basic business activities and etiquette.

Level 2:  Topics for study include, arranging company meetings and presenting product information. We will also familiarise you with the basic structure of Chinese characters.

Level 3: You will learn about business proposals and sales negotiations.

Level 4: You will study business situations related to placing simple orders.

Level 5: Sales strategies, product improvements and understanding more complex business documentation.

Level 6: The introduction to financial services, intellectual property and economic trends feature in our Level 6 material.

Level 7: Product planning and launch together with marketing tactics are covered during level 7.

Level 8: E-commerce, supply chain management and business schemes are discussed in Level 8.

Level 9: By this level, students should be prepared to appreciate and interpret practically all forms of the written business Chinese language and be able to speak to near native fluency. We will expect you to find the underlying meaning in business texts and write a thesis related to your findings .

Travel Chinese

The aims of the course are to familiarize students with the Chinese names of popular places in China, have an adequate command of the language in order to travel, conduct basic conversational techniques (e.g. asking directions, ordering food, shopping).

For our advanced students, we look to teach you the more sophisticated phrases and dialogues you can use whilst travelling in China. This includes understanding Chinese slang and equipping you for complex situations like bargaining with the locals.    

Chinese Culture

This course is really about broadening your views and knowledge about China by exploring the history, traditions, customs and arts of the country.

 

The Inverse of A Matrix

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Let A  be an n*n  matrix. An n*n  matrix A-1  such that                                                                                      AA-1=A-1A=In is the inverse of A. A matrix A  is nonsingular if A-1exists (i.e., if A has an inverse). If a matrix does not have an inverse, then it is singular. Examples Explanation                 Calculating The Inverse Given a square matrix M, we know the size of its inverse (the same size as M) and the product of M  and its inverse. Using this information, the inverse of M  can be calculated by assigning variables to the elements of M -1 and representing the product MM -1 as a system of n  equations in n  unknowns, where M  has dimension n*n. Examples Explanation     This system can be expressed as an augmented matrix. Recall that an augmented matrix includes the constant coefficients of a system of equations.     The solution to this system will provide the elements of M -1.   Solving Systems of Equations with The Matrix Inverse Systems of linear equations can be solved using the matrix inverse. The system must be represented by the equation                                                                                               AX=B where A  is the coefficient matrix of the system, X  is the column matrix of variables, and B  is the column matrix of constant coefficients. The solution to the system will then be A -1B  because                                                                                    Examples Explanation         The solution is (x,y)=(3,4)   Try these exercises Solve 1. Show that 2. Show that the inverse of M  is 3. Show that 4. Find the inverse of N  using a system of four equations in four unknowns. 5. Find the inverse of N  using a system of four equations in four unknowns. 6. Is the identity matrix singular or nonsingular? 7. Solve the system  using elementary row operations and matrix inverses. 8. Solve the system using elementary row orations and matrix inverses. 9. Solve the system using elementary row operations and matrix inverses. 10. Is it possible to solve a system composed of less unknowns than equations using elementary row operations and the matrix inverse? Answers to questions 1. 2.                                  3.       4. 5.                                                                                                                                    By substitution, 6. No elementary row operations are required to transform In  into In, so In-1=In. Since In has an inverse, it is nonsingular. 7. 8. 9.    . 10. Yes, but the extraneous equations must be eliminated so that the coefficient matrix is square (having the same number of equations as unknowns).