1.1 Some useful properties

1.2 Basic math operations

\[\begin{matrix} + & Combining\\ - & Taking\ away\\ \times & Adding\ repeatedly\\ \div & Splitting\ up\\ \end{matrix}\]

1.3 Clue words

1.4 Example of clue words

1.5 Sample Problem 1

In retailing, the difference between the gross sales and customer returns and allowances is called the net sales. If a store’s gross sales were $2,538 and customer returns and allowances amounted to $388, what was the store’s net sales?

1.6 Sample Problem 2

The population of the United States in 1800 was 5,308,483. Ten years later, the population had grown to 7,239,881. During this period of time, did the country’s population double?

2.1 Properties of the equality \(a=b\)

4.1 Example of an Algebratic expression

\[\frac{x+2}{4} = \frac{x-1}{3} +2\]

\[\frac{2x+8}{x} = \frac{30}{5}\]

4.2 Solution

\[\eqalign{\frac{x+2}{4} &=& \frac{x-1}{3} +2\\ \color{red}{12}\left(\frac{x+2}{4}\right) &=& \color{red}{12}\left(\frac{x-1}{3} +2\right)\\ 3\left(x+2\right)&=&4\left(x-1\right) + 24\\ \color{red}{-4x -6} +3x + 6 &=& 4x -4 + 24 + \color{red}{-4x -6}\\ \color{red}{(-1)}-x &=& 14 \times \color{red}{(-1)}\\ x &=& -14\\ }\]

4.3 Solution

\[\eqalign{\frac{2x+8}{x} &=& \frac{30}{5}\\ \color{red}{x}\left(\frac{2x+8}{x}\right) &=& \color{red}{x}\left(6\right)\\ \color{red}{-2x}+2x+8 &=& 6x +\color{red}{-2x}\\ \color{red}{\frac{1}{4}} \times 8 &=& 4x \times \color{red}{\frac{1}{4}}\\ 2 &=& x\\ }\]

4.4 Examples of an Algebratic expression

\[\frac{x+2}{4} = \frac{x-1}{3} +2\]

5.1 Solution 1

\[\eqalign{2(x-3) - 17 &=& 13 - 3(x+2)\\ 2x-23 =2x -6 -17 &=& 13-3x-6 = -3x+7\\ 2x-23\color{red}{+3x+23} &=& -3x+7\color{red}{+3x+23}\\ \color{red}{\frac{1}{5}}5x &=& 30\color{red}{\frac{1}{5}}\\ x &=& 6\\ }\]

5.2 Solution 2

\[\eqalign{\frac{3}{x+6} + \frac{1}{x-2} &=& \frac{4}{4x -16}=\frac{1}{x-4}\\ \color{red}{\frac{(x+6)(x-2)}{(x+6)(x-2)}}\left(\frac{3}{x+6} + \frac{1}{x-2}\right)&=& \frac{1}{x-4}\\ \frac{3(x-2)+(x+6)}{(x+6)(x-2)}&=& \frac{1}{x-4}\\ \frac{4x}{(x+6)(x-2)}=\frac{3x-6+x+6}{(x+6)(x-2)}&=& \frac{1}{x-4}\\ \color{red}{(x+6)(x-2)(x-4)}\frac{4x}{(x+6)(x-2)}&=& \frac{1}{x-4}\color{red}{(x+6)(x-2)(x-4)}\\ 4x(x-4)&=& (x+6)(x-2)\\ 4x^2 +16x\color{red}{-(x^2+4x -12)}&=& x^2 +4x-12\color{red}{-(x^2+4x -12)}\\ 3(x^2 + 4x + 4)= 3x^2 + 12x +12 &=& 0\\ (x+2)^2 &=& 0\\ x = -2 }\]

5.3 Solution 3

\[\eqalign{\frac{1}{x} + \frac{1}{q} &=& \frac{1}{f}\\ \frac{1}{x} + \frac{1}{q} + \color{red}{\frac{-1}{q}}&=& \frac{1}{f} - \color{red}{\frac{1}{q}}\\ \frac{1}{x} &=& \left(\frac{1}{f} - \frac{1}{q}\right)\color{red}{\frac{fq}{fq}}=\frac{q-f}{qf}\\ x &=& \frac{qf}{q+f}\\ }\]

5.4 Solution 4

\[\eqalign{11(x + 5) &=& 25(x-5)\\ \color{red}{+125 -11x} +11x + 55 &= & 25x -125\color{red}{+125 -11x}\\ \color{red}{\frac{1}{14}}180 &=& 14x\color{red}{\frac{1}{14}}\\ \frac{90}{7} = \frac{180}{14}&=&x\\}\]

5.5 Multiplication of polynomials

\[\eqalign{\color{red}{(a+b)}(c+d)&=& \color{red}{a}(c+d) \color{red}{+ b}(c+d)\\ &=& ac + ad + bc + bd\\}\]

\[\eqalign{\color{red}{(ax+b)}(cx+d) &=& \color{red}{ax}(cx + d) \color{red}{+ b}(cx + d)\\ &=& acx^2 +adx + bcx + bd\\ &=& acx^2 + \left(ad+bc\right)x +bd\\}\]

5.6 Examples

\[\eqalign{\color{red}{(x+3)}(x-2) &=& \color{red}{x}(x-2)\color{red}{+3}(x-2)\\ &=& (x^2 -2x) + (3x - 6)\\ &=& x^2 + x -6\\}\]

6.1 Difference of squares

\[\eqalign{\color{red}{(x+3)}(x-3) &=& \color{red}{x}(x-3)\color{red}{+3}(x+3)\\ &=&(x^2 -3x) + (3x - 9)\\ &=&(x^2 +(-3x + 3x) - 9)\\ &=&x^2 - 9\\}\]

7.1 Definitions:

• Dividends: a specific amount of money given to participents or sponsers on a regular basis, e.g. 200 baht per quarter. If P= Principle, D=Dividend: \(P_{i+1} = P_i + D\)

• Interest: regular money paid back to participant based on a specific rate, e.g. 2% per annum. If P= Principle, I=Interest rate: \(P_{i+1} = P_i + P_i \times I\)

• Tax: amount owed to the government upon receipt of the funds. Based on a percentage of the interest accrued.

7.2 Report format

7.3 Solving Word Problems

• Read the problem carefully.
• Choose a strategy.
• Decide which basic operation(s) are relevant and then translate the words into mathematical symbols.
• Perform the operations.
• Check the solution to see if the answer is reasonable.

7.4 Sample Problem 1

In retailing, the difference between the gross sales and customer returns and allowances is called the net sales. If a store’s gross sales were $2,538 and customer returns and allowances amounted to $388, what was the store’s net sales?

7.5 Sample Problem 2

The population of the United States in 1800 was 5,308,483. Ten years later, the population had grown to 7,239,881. During this period of time, did the country’s population double?

7.6 Sample Problem 3.

A delivery van travels 27 miles west, 31 miles east, 45 miles west, and 14 miles east. How far is the van from its starting point?

7.7 Sample Problem 4.

Recycling one aluminum can saves enough energy to run a television for three hours. The average American watches 3,048 hours of television a year. For a year, how many aluminum cans would it take to power a television for the average American?

7.8 Sample Problem 5.

A blue whale weighs about 300,000 pounds, and a great white shark weighs about 4,000 pounds. How many times the weight of a great white shark is the weight of a blue whale?

7.9 Sample Problem 6.

A sales representative flew from Los Angeles to Miami (2,339 miles), then to New York (1,092 miles), and finally back to LA (2,451 miles). How many total miles did he fly?

7.10 Sample Problem 7.

A movie fan installed shelves for his collection of 400 DVDs. If 36 DVDs fit on each shelf, how many shelves did he need to house his entire collection?

7.11 Sample Problem 8.

Two major naval disasters of the twentieth century involved the sinking of British ships—the Titanic and the Lusitania. The Titanic, which weighed about 93,000,000 pounds, was the most luxurious liner of its time; it struck an iceberg on its maiden voyage in 1912. The Lusitania, which weighed about 63,000,000 pounds, was sunk by a German submarine in 1915. How much heavier was the Titanic than the Lusitania?

8.1 Linear with offset

\(y = mx + b\)

\(F = 32 + \frac{9}{5} C\)

8.2 Crickets chirps

8.3 Global warming.

Scientists suspect hat rising sea temperatures will have an adverse effect on coral growth. Here is the results of a study:

8.4 What is wrong with the following phrases?

• Most generic medications aren’t 50% or 75% less expensive than their brand named equivalents, they are 100 times cheaper!!

• Generic medications can cost 100 percent less than their brand-name equivalent

• I love her with 200% of my heart.

8.5 NASA Challenger O-ring problems

8.6 Exponenial Growth

8.7 Exponential growth

8.8 Simultaneous linear equations

8.9 Simultaneous equations Part 2

8.10 Practice

\[\eqalign{ 10x + 5y &=& 30\\ 2x + 12y &=& 40\\ \hline 2x + 3y &=& 15\\ 5x + y &=& 10\\ \hline x + 5y &=& 20\\ 3x + 15y &=& 60\\ }\]

8.11 A word about units

If the formula is correct, the units should match.

Example

The typical cloud contains 1,100,000,000 pounds of water. How many litres of rain does that represent?

\[1.1\times 10^9 lbs \times \frac{1 kg}{2.2 lb} \times \frac{1 L}{1 kg}= 5 \times 10^8 L\]