Inequalities

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課程筆記

What is an inequality?

An inequality compares two expressions that are not equal, using symbols like <, >, ≤, ≥, or ≠.
Inequalities are often used to compare numbers by their size on the number line.
The main types are less than (<) and greater than (>).
For example, a b means a is greater than b.

Inequality symbols

Strict inequalities use < or > and exclude equality: a < b means a is strictly less than b.
Non-strict inequalities use ≤ or ≥ and include equality: a ≤ b means a is less than or equal to b (at most b).
Similarly, a ≥ b means a is greater than or equal to b (at least b).
The symbol ≠ means not equal to; it does not say which is larger.
The symbols < and > are symmetrical: a a.

Representing inequalities on a number line

An inequality in one variable can be shown on a number line by shading the region that satisfies it.
Use an open circle for strict inequalities (< or >) to show the endpoint is not included.
Use a closed circle for non-strict inequalities (≤ or ≥) to show the endpoint is included.
For example, x > 3 is shown with an open circle at 3 and shading to the right.
x ≤ −1 is shown with a closed circle at −1 and shading to the left.

Listing integer solutions

Integer values that satisfy an inequality can be listed by testing whole numbers in the range.
For x > 2, the integer solutions are 3, 4, 5, … (2 is not included).
For x ≤ 4, the integer solutions are 4, 3, 2, 1, 0, −1, … (4 is included).
For −1 < x ≤ 3, the integer solutions are 0, 1, 2, 3.
Always check whether the endpoints are included (closed circle) or excluded (open circle).

Solving linear inequalities

Solve linear inequalities in one variable using the same steps as solving equations: add, subtract, multiply, or divide both sides.
Addition and subtraction: adding or subtracting the same number on both sides keeps the inequality true.
Multiplication and division by a positive number: the inequality sign stays the same.
Multiplication and division by a negative number: the inequality sign reverses (e.g., < becomes >).
For example, to solve −2x > 6, divide both sides by −2 and reverse the sign: x < −3.
After solving, represent the solution on a number line or list integer values if asked.

Key properties of inequalities

Transitivity: if a < b and b < c, then a < c.
Converse: ≤ and ≥ are converses; a ≤ b is equivalent to b ≥ a.
Additive inverse: if a < b, then −a > −b.
Multiplicative inverse: if a and b are both positive and a < b, then 1/a > 1/b.
Applying a monotonically increasing function (like adding a constant) preserves the inequality.
Applying a monotonically decreasing function (like multiplying by a negative) reverses the inequality.

Common mistakes to avoid

Forgetting to reverse the inequality sign when multiplying or dividing by a negative number.
Using an open circle when the endpoint should be included (≤ or ≥).
Including the endpoint when the inequality is strict (< or >).
Treating ≠ as if it tells you which number is larger — it does not.

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練習題

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1.Which symbol means 'is strictly less than'?
Easy
- A<
- B>
- C≤
- D≥
2.Which of the following is a non-strict inequality?
Easy
- Aa b
- Ca ≤ b
- Da ≠ b
3.The inequality a > b means that a is greater than b and a is not equal to b.
Easy
True or false?
4.If x > 3, which of these is the smallest integer that satisfies the inequality?
Easy
- A2
- B3
- C4
- D5
5.If y ≤ 5, which of these integers does NOT satisfy the inequality?
Easy
- A3
- B4
- C5
- D6
6.Solve the inequality −2x ≥ 8.
Medium
- Ax ≤ −4
- Bx ≥ −4
- Cx ≤ 4
- Dx ≥ 4
7.Match each inequality statement with its meaning in words.
Medium
- a b
- a ≤ b
- a ≥ b
- a is less than b
- a is greater than b
- a is less than or equal to b
- a is greater than or equal to b
8.Order these numbers from smallest to largest.
Medium
- -5
- -1
- 0
- 2
- 4

複習筆記

## Inequalities — KS3 Algebra An **inequality** compares two expressions that are **not equal**, usually by size on the number line. ### Symbols - a b means a is **greater than** b. - a ≤ b means a is **less than or equal to** b (at most b). - a ≥ b means a is **greater than or equal to** b (at least b). < and > are **strict** inequalities — equality is excluded. ≤ and ≥ are **non-strict** — equality is allowed. Also: a ≠ b means a is **not equal to** b. Mirroring works: a a are equivalent. ### Number lines - **Open circle** = strict (< or >): that value is **not** included. - **Closed (filled) circle** = non-strict (≤ or ≥): that value **is** included. - Shade the arrow in the direction of the solutions. ### Listing integer values For −2 < x ≤ 3, the integers are −1, 0, 1, 2, 3. (Open circle excludes −2; closed circle includes 3.) ### Solving linear inequalities Solve like an equation, but remember: - Adding or subtracting the same constant on both sides **keeps** the inequality. - Multiplying or dividing both sides by a **positive** number **keeps** the inequality. - Multiplying or dividing both sides by a **negative** number **reverses** the sign (< becomes >, ≤ becomes ≥). Example: −3x < 12 → divide by −3 and reverse → x > −4. ### Key properties - **Transitivity:** if a −b. ### Exam tips 1. Show the circle type clearly on the number line. 2. Check your answer by testing a value. 3. Never forget to reverse the sign when dividing by a negative.

Inequality: A relation making a non-equal comparison between two numbers or expressions, usually by size on the number line.
Strict inequality: An inequality using < or >, where equality is excluded.
Non-strict inequality: An inequality using ≤ or ≥, where equality is allowed.
a < b: a is less than b.
a > b: a is greater than b.
a ≤ b: a is less than or equal to b (at most b).
a ≥ b: a is greater than or equal to b (at least b).
a ≠ b: a is not equal to b.
Open circle: On a number line, shows the endpoint is not included (used for < or >).
Closed circle: On a number line, shows the endpoint is included (used for ≤ or ≥).
Reversing the sign: When multiplying or dividing an inequality by a negative number, the inequality symbol flips direction.
Transitivity: If a < b and b < c, then a < c.

字卡

What does the inequality symbol < mean?

a < b means a is less than b. It is a strict inequality, so a is not equal to b.

What does the inequality symbol > mean?

a > b means a is greater than b. It is a strict inequality, so a is not equal to b.

What does the inequality symbol ≤ mean?

a ≤ b means a is less than or equal to b, i.e. a is at most b. It is a non-strict inequality.

What does the inequality symbol ≥ mean?

a ≥ b means a is greater than or equal to b, i.e. a is at least b. It is a non-strict inequality.

What does the inequality symbol ≠ mean?

a ≠ b means a is not equal to b. It does not say which is greater, and it does not require a and b to be in an ordered set.

On a number line, what does an open circle represent?

An open circle shows that the endpoint value is not included in the solution set. It is used for strict inequalities < and >.

On a number line, what does a closed (filled) circle represent?

A closed circle shows that the endpoint value is included in the solution set. It is used for non-strict inequalities ≤ and ≥.

How do you show the inequality x > 3 on a number line?

Draw an open circle at 3 and shade or draw an arrow to the right, because x is greater than 3 but not equal to 3.

How do you show the inequality x ≤ -1 on a number line?

Draw a closed circle at -1 and shade or draw an arrow to the left, because x is less than or equal to -1.

List the integer values that satisfy x < 4.

The integers less than 4 are ..., 1, 2, 3. (If restricted to positive integers, the list is 1, 2, 3.)

List the integer values that satisfy -2 ≤ x < 3.

The integers from -2 up to but not including 3 are -2, -1, 0, 1, 2.

What happens to an inequality when you add or subtract the same number on both sides?

The inequality sign stays the same. For example, if a < b then a + c < b + c and a - c < b - c.

What happens to an inequality when you multiply or divide both sides by a positive number?

The inequality sign stays the same. For example, if a 0, then ac < bc and a/c < b/c.

What happens to an inequality when you multiply or divide both sides by a negative number?

The inequality sign reverses. For example, if a bc and a/c > b/c.

Solve the inequality x + 5 > 2.

Subtract 5 from both sides: x > -3. The sign does not change because you only subtracted.

Solve the inequality 3x ≤ 12.

Divide both sides by 3 (positive): x ≤ 4. The sign stays the same.

Solve the inequality -2x < 8.

Divide both sides by -2 (negative), so reverse the sign: x > -4.

Solve the inequality 4 - x ≥ 1.

Subtract 4: -x ≥ -3. Multiply or divide by -1 and reverse the sign: x ≤ 3.

Are a a equivalent?

Yes. The two symbols mirror each other, so a a mean the same thing.

What does the transitive property of inequality state?

If a < b and b < c, then a < c. If either premise is strict, the conclusion is strict.

What does a ≪ b mean?

a is much less than b, normally by several orders of magnitude. The lesser value can often be neglected in approximations.

What does a ≫ b mean?

a is much greater than b, normally by several orders of magnitude. The lesser value can often be neglected in approximations.

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