IT103: Algebra

Unit 2: Equations and Inequalities

R Batzinger

2024-07-12

2.1 The Rectangular Coordinate Systems and Graphs

2.2 Linear Equations in One Variable

2.3 Models and Applications

The basic formula for a line:

\[y=mx+b\] * \(m\) = Slope

\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

  • \(b\) = y intercept

Identical slopes

Identical intercepts

Solution space

Practical problem

Caroline is a full-time college student planning to go home during spring break. To earn enough money for the trip, she has taken a part-time job at the local bank that pays $15.00/hr, and she opened a savings account with an initial deposit of $400 on January 15. She arranged for direct deposit of her payroll checks.

  • If spring break begins March 20 and the trip will cost approximately $2,500, how many hours will she have to work to earn enough to pay for her vacation?

  • If she works 4 hours per day, how many days to reach her goal?

Equivalent fractions (Cross products)

\[\eqalign{\frac{a}{b} &=& \frac{c}{d}\\ \frac{a\color{red}{\cdot b \cdot d}}{b} &=& \frac{c\color{red}{\cdot b \cdot d}}{d}\\ a\cdot d &=&b \cdot c\\}\]

Scaling a recipe

Amount Ingredient Abbrev
230gm butter B
200gm sugar S
4 eggs E
125gm flour F
4gm baking powder P

  • Standard batch

\[\tiny\frac{1}{1} =\frac{230}{B}= \frac{200}{S}=\frac{4}{E}=\frac{125}{F}=\frac{4}{P}\] * Doubled Batch

\[\tiny\eqalign{\frac{2}{1}&=& \frac{2\times230}{B}&= \frac{2\times200}{S}&= \frac{2\times 4}{E}&= \frac{2\times125}{F}&= \frac{2\times4}{P}\\ \frac{2}{1}&=& \frac{460}{B}&= \frac{400}{S}&= \frac{8}{E}&= \frac{250}{F}&= \frac{8}{P}\\}\]

  • Halved batch

\[\tiny\eqalign{\frac{\frac{1}{2}}{1}&=& \frac{\frac{1}{2}\times230}{B}&= \frac{\frac{1}{2}\times200}{S}&= \frac{\frac{1}{2}\times 4}{E}&= \frac{\frac{1}{2}\times125}{F}&= \frac{\frac{1}{2}\times4}{P}\\ \frac{\frac{1}{2}}{1}&=& \frac{115}{B}&= \frac{100}{S}&= \frac{2}{E}&= \frac{62.5}{F}&= \frac{2}{P}\\}\]

General principles of Modelling

Description Math equivalent
Multiply by \(a\) \(a\cdots x\)
Scaled by \(a\) \(a\cdots x\)
Offset by \(b\) \(x + b\)
Start with \(b\) and add \(2x\) \(b + 2x\)

Example

  • Cost of Car rental 200 baht per day and 20 baht per km

\[C = 200 D + 20\cdot K\]

  • Monthly Water bill 40 baht plus 50 baht per KL

\[W= 40 + 50K\] * Glide distance is the initial altitude minus the descent rate over time

\[h=A-d\cdot t\]

2.4 Complex Numbers

Adding complext numbers

  • Add real terms
  • Add imaginary terms
  • Combine

\[(3+5i) + (2+4i) = 5 + 9i\]

Multiplying a Complex Number by a Real Number

  • Use the distributive property.
  • Simplify.

\[3(4 +2i) = 3\cdot 4 + 3 \cdot 2i = 12 + 6i\]

Multiply 2 Complex Numbers

  • Use the distributive property.
  • Simplify.

\[\eqalign{4 +2i\times 5+3i &=& 4\cdot 5 + 4\cdot 3i + 2i\cdot 5 + 3\cdot2\cdot i^2\\ &=& 20 + 12i + 10i -6\\ &=& 14 + 22i\\}\]

2.5 Quadratic Formula

\[ax^2 + bx +c = 0\] \[x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}\]

Example

\[x^2 -9x +20= (x-4)(x-5) =0\] \[\eqalign{x&=& \frac{-(-9)\pm \sqrt{-9^2 - 4\cdot20}}{2\cdot(1)}\\ &=& \frac{9\pm \sqrt{81 - 80}}{2}\\ &=&\frac{9\pm 1}{2}\\ &-&\frac{(8,10)}{2}\\ &=&(4,5)\\}\]

Another example

\[4x^2 -16x +8= 0\] \[\eqalign{x&=&\frac{-16\pm\sqrt{-16^2-4\cdot4\cdot8}}{2\cdot 4}\\ &=&\frac{-16\pm\sqrt{256-128}}{8}\\ &=&\frac{-16\pm\sqrt{128}}{8}\\ &=&\frac{-16\pm11.3137}{8}\\ &=&\frac{(-16+11.3137,-16-11.3137)}{8}\\ &=&\frac{(-4.6863,-27.3137)}{8}\\ &=&(0.58579,3.41421)\\}\]

2.7 Linear Inequalities

  • \(x < 5\) x less than 5
  • \(x \le 5\) x less than or equal 5
  • \(x = 5\) x equals 5
  • \(x \ge 5\) x greater than or equal 5
  • \(x > 5\) x greater than 5

Properties of inequalities

Given \(a<b\)

  • Addition

\[a + c < b + c\] * Multiplication

\[if (a < b) \land (c > 0):\quad ac < bc\]

\[if (a < b) \land (c < 0):\quad ac > bc\]

Absolute value

\[if\ (x > 0): \quad |x| = x\] \[if\ (x< 0): |x| = - x\]

Simplifying inequality

\[\eqalign{-x +4 &\le& \frac{x}{2} +1\\ -\frac{x}{2}&\le& -3\\ x&\ge& 6\\}\]

Two side inequalities

\[\eqalign{4 &\lt& 2x -8 &\le& 10\\ 12 &\lt& 2x &\le&18\\ 6 &\lt& x &\le& 9}\]

Another example

\[|x -1| \le3\] \[|x| \le 4\] \[-x\le4;\quad x \le 4\]

\[x\ge-4;\quad x \le4\] \[-4 \le x \le4\]

IT103 Algebra