Section 01
Inequality Basics
Understanding inequality symbols, their meaning, and how to read them on a number line.
⚡ Quick Memory — Key Vocab
The 4 Inequality Signs
$a < b$ → a is LESS than b
$a > b$ → a is GREATER than b
$a \leq b$ → LESS or EQUAL
$a \geq b$ → GREATER or EQUAL
OPEN dot = strict ( $<$ or $>$ )
CLOSED dot = includes ( $\leq$ or $\geq$ )
📘 Worked Example
Graph the solution of $x > 3$ on a number line. What kind of dot do you use at 3?
1
The symbol is $>$ (strict greater than), so 3 is NOT included.
2
Draw an open circle at 3, then shade everything to the right (toward $+\infty$).
3
Memory trick: open = strict = hollow dot
1
Which of the following correctly describes $-3 \leq x$?
2
On a number line, the solution $x < 5$ is shown with:
Section 02
One-Step Inequalities
Add, subtract, multiply, or divide — but remember the golden rule about negatives.
🔴 Critical Rule — Most Common Mistake
Multiplying / Dividing by a NEGATIVE → FLIP the sign!
$-2x > 6$ → divide by $-2$ → FLIP → $x < -3$
Add / Subtract → sign stays SAME
× or ÷ positive → sign stays SAME
× or ÷ negative → FLIP!
📘 Worked Example
Solve: $-\dfrac{n}{3} < -1$
1
Multiply both sides by $-3$ (negative!) → must FLIP the sign.
2
$$n > (-1) \times (-3) = 3$$
3
Answer: $n > 3$ — open dot at 3, shade right.
3
Solve: $3w \geq 6$
4
Solve: $-30 > -5h$
5
Solve: $-\dfrac{n}{3} < -1$. What is the correct solution?
Section 03
Multi-Step & Integer Solutions
Combine like terms, use distributive property, then find the smallest or largest integer.
🟢 Key Phrase Memory
Smallest Integer / Largest Integer
"smallest integer": solve → round UP to nearest whole
"largest integer": solve → round DOWN to nearest whole
$x \geq 3.2$ → smallest integer = 4
$x \leq 3.8$ → largest integer = 3
📘 Worked Example
Find the smallest integer $x$ satisfying $4x \geq 16$.
1
Divide both sides by 4 (positive, no flip): $x \geq 4$
2
Integers that satisfy $x \geq 4$: $4, 5, 6, \ldots$
3
Smallest integer = 4
6
Find the smallest integer $x$ satisfying $-\dfrac{6}{5} < \dfrac{3}{2}x$.
7
Solve: $2(x - 3) + 5 > 3$. What is the solution?
8
Solve: $5 - 3x \leq 14$
Section 04
Properties & Proofs
Addition, subtraction, multiplication properties of inequality — and how to use them logically.
🟣 Proof Toolkit
5 Properties of Inequality
Addition: $a < b \Rightarrow a+c < b+c$
Subtraction: $a < b \Rightarrow a-c < b-c$
× positive: $a < b, c>0 \Rightarrow ac < bc$
× negative: $a < b, c<0 \Rightarrow ac > bc$ (FLIP!)
Transitivity: $a < b, b < c \Rightarrow a < c$
9
Let $k < 0$. If $a < b$, which must be true?
10
If $a < b$ and $b < c$, what can we conclude by transitivity?
Section 05
Word Problems & Tricky Cases 🌍 Real World
Translate English into math — the most common place to lose marks. Read carefully!
🟠 Word → Math Decoder
Key English Phrases for Inequalities
"at least" → $\geq$
"at most" → $\leq$
"no more than" → $\leq$
"more than / exceeds" → $>$
"fewer than / less than" → $<$
"minimum" → $\geq$
"maximum" → $\leq$
📘 Worked Example — Word Problem
Juno needs to earn at least $500 this week. She already earned $175. She earns $25 per hour. How many hours $h$ must she work?
1
"At least $500" → total $\geq 500$
2
Total = $175 + 25h$, so: $175 + 25h \geq 500$
3
$25h \geq 325 \Rightarrow h \geq 13$. She must work at least 13 hours.
11
A box can hold at most 50 kg. It already contains items weighing 32 kg. You want to add $x$ more kg. Which inequality represents this situation?
12
Mia scores 72, 85, and 91 on three tests. She needs an average of at least 80 over 4 tests. What is the minimum score she needs on the 4th test?
13
A parking garage charges ₩2,000 flat fee plus ₩500 per 30-min. You have ₩7,000. What is the maximum number of full 30-minute intervals you can park?
14
If $a < b$ and both $a$ and $b$ are positive, what is the relationship between $\dfrac{1}{a}$ and $\dfrac{1}{b}$?
15
If $a < b < 0$ (both negative), what is the relationship between $\dfrac{1}{a}$ and $\dfrac{1}{b}$?
16
A store offers two plans for buying coffee:
Plan A: ₩3,000 per cup
Plan B: ₩15,000 monthly membership + ₩1,500 per cup
After how many cups does Plan B become cheaper?
Plan A: ₩3,000 per cup
Plan B: ₩15,000 monthly membership + ₩1,500 per cup
After how many cups does Plan B become cheaper?
17
Solve: $\dfrac{2x - 1}{3} \geq x - 2$
18
Tom is 3 times as old as Jerry. In 5 years, Tom's age will be more than twice Jerry's age. What is the minimum current age of Jerry if Jerry's age is a whole number?
19
Which of the following is a correct step when solving $\dfrac{x}{-4} > 3$?
20
A rectangle has a width of $(2x - 1)$ cm and a length of $(x + 4)$ cm. Its perimeter must be less than 38 cm. What is the range of valid values of $x$? (Assume $x > 0$)
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