02_Equations_and_Inequalities

2.1 The Rectangular Coordinate Systems and Graphs

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2.2 Linear Equations in One Variable

• a linear equation is an equation of a strait line, written in one variable. It can be written in the form

$$ax + b = 0$$

​ where a and b are real numbers, $a \neq 0$

linear equations in one variable can be classified into three types:

• identity equation: true for all values on the variable

  $3x = 2x + x$

• conditional equation: true for only some values of the variable

  $$\begin{align*} 5x + 2 &= 3x -6 \\ 2x &= -8 \\ x &= -4 \end{align*}$$
• inconsistent equation: is a false statement

  $5x - 15 = 5x - 20$

2.4 Complex Numbers

• a complex number is the sum of a real number and an imaginary number

a complex number is a number of the form $a + bi$, if $a = 0$ and $b \neq 0$, the complex number is called a pure imaginary number

• an imaginary number is an even root of a negative number

2.5 Quadratic Equations

zero-product property states:

if $a \cdot b = 0$, then a = 0 or b = 0

a quadratic equation is an equation containing a second-degree polynomial; e.g.

$$ax^2 + bx + c = 0$$

where a, b, and c are real numbers, and $a \neq 0$