Module 1 Part 2 Linear Inequalities

2026-02-28 19:11

Tags: #math

Author: Duke Hsu

Module 1 Part 2 - Linear Inequalities

Single Variable

inequality

inequality: An inequality is like an equation, but instead of an equal sign (=), is has one of these signs:

SYMBOL	MEANING
\(<\)	less than
\(\le\)	less than or equal to
\(>\)	greater than
\(\ge\)	greater than or equal to

Graphing

number line

Example:

Graphing Rules (number line):

SYMBOL	CIRCLE	DIRECTION OF ARROW
\(<\)	open	left
\(>\)	open	right
\(\le\)	closed	left
\(\ge\)	closed	right

Interval Notation

SYMBOL	MEANING	READING
\((\quad)\)	endpoint NOT included	Parentheses
\([\quad]\)	endpoint included	Brackets
\(\infty \quad -\infty\)	always use in parentheses	infinity

Set Builder Notation

use curly braces { }

Formart: \({x \quad| \quad condition}\)

Meaning:

The set of all \(x\) such that the conditions is true .

Example:

\(x > 3\), \(\{x | x > 3\}\)

Example

Question 1:

\(-3x +4 <13\)

Note

兩邊都減去4 ，然後再除以-3

(因為9 除以負數，所以要轉換符號)

\(x >-3\)

Step1-Number line

Step2-Interval Notation

\((-3,\infty )\)

Step3- Set Builder Notation

\(\{x\quad| \quad x > -3 \}\)

Value of \(x\): \(x=-2,-1,0,1,2,3,4,5...\)

Question 2:

\(-5 \le x \le -1\)

Step1 - Number line

Step2- Interval Notation

\([-5,-1]\)

Step3- Set Builder Notation

\(\{x | -5\le x \le -1\}\)

Values of \(x\) : \(x=-5,-4,-3,-2,-1\)

Question 3:

\(x < -3 \quad or \quad x > 3\)

Step1 Number line

Step2 Interval Notation

\((- \infty , -3) \quad (3, \infty)\)

Step3 Set Builder Notation

\(\{x\quad | \quad x<-3\}\quad U \quad \{x\quad |\quad x>3\}\)

Values of \(x\):

\(x= -4,-5,-6,-7,-8,-9.....\) \(x=4,5,6,7,8,9,10\)

Question 4:

\(x\le -5 \quad or\quad x>10\)

Step1 Number line

Step2 Interval Notation

\((-\infty, -5]\) \((10,\infty)\)

Step3 Set Builder Notation

\(\{x\quad | \quad x\le-5\}\quad U \quad \{x\quad |\quad x>10\}\)

Values of \(x\):

\(x=-5,-6,-7,-8,....\)

\(x=11,12,13,14,15\)

Question 5:

\(5+x\ge7\)

\(x\ge 2\)

Step1 Number line

Step2 Interval Notation

\([2,\infty)\)

Step3 Set Builder Notation

\(\{x \quad | \quad x \ge 2\}\)

Values of x: \(x=2,3,4,5,6,7,8,9.....\)

Question 6:

\(\frac{x}{3} \ge 5\)

Multiply both side by : 3 * (x/3) >= 5 * 3

left side x, right side 15,

Result: \(x \ge 15\)

Step1: Number line

Step2: Interval Notation

\([15, \infty)\)

Step3: Set Builder Notation

\(\{x\quad |\quad x \ge15\}\)

Values of x: \(x= 15,16,17,18,19,20,21,.....\)

Question 7:

\(-2\le x+1 \le 4\)

Calculator \(x\):

\(\because\) \((-2)-(-1) \le x \le 4-1\)

\(\therefore\) \(-3\le x \le 3\)

Step1 Number line

Step2 Interval Notation

\([-3,3]\)

Step3 Set Builder Notation

\(\{x| -2\le x+1 \le4\}\)

Values of \(x\):

\(x=-3,-2,-1,0,1,2,3\)

Solving it in Two variable

Quadrant Graph

Two types of Line

LINE NAME	EXAMPLE	SYMBOL
Dotted line	--------	\(>,<\)
Solid line	———	\(\ge,\le\)

Example:

Note

垂直線(Vertical Line)： inequality 中只有x , 沒有y；

Note

水平線 (Horizontal Line): inequality 中只有y, 沒有x；

Note

斜線 (Slanted/Oblique Line) inequality 中有y，也有x；

Graph the inequality

Question : \(3x-5y\ge 15\)

Step 1. Replace inequality symbol with equals sign: \(3x – 5y = 15\)

Step 2. Graph the line

Calculator the point:

x	y
0	-3
5	0

point A: (0,-3) point B: (5,0)

Since: \(\ge\) , therefor is Solid Line, and in QIV,

如何判斷二元一次不等式的區域（Range）（如何畫出陰影）

當你畫出一條直線（例如 \(2x + 4y = 8\)）後，這條線把坐標平面分成了兩半。要判斷該塗哪一邊，最簡單的方法是「原點測試法」。

式子： \(2x + 4y \leq 8\)
步驟：
1. 先畫出直線 \(2x + 4y = 8\)（通過 \((4, 0)\) 和 \((0, 2)\)）。
2. 找一個不在線上的點代入。最方便的就是原點 \((0, 0)\)。
3. 將 \(x=0, y=0\) 代入不等式：
  
  \[2(0) + 4(0) \leq 8 \implies 0 \leq 8\]
4. 判斷： \(0 \leq 8\) 是成立的。這代表原點所在的區域就是「正確的區域」。
5. 結論： 塗向包含 \((0, 0)\) 的那一側（即直線的左下方）。

References

Module 1 PDF

Gemini tools - https://gemini.google.com/share/b3c0ec6c89db

Module 1 Part 2 Linear Inequalities