Inequalities

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Interpreting Inequalities

An inequality compares two values: > (greater than), < (less than), ≥ (greater than or equal to), ≤ (less than or equal to).
Strict inequalities use > or < and do not include the endpoint.
Non‑strict inequalities use ≥ or ≤ and do include the endpoint.
Example: x > 5 means x can be 6, 7, 8, … (not 5).
Example: x ≤ 10 means x can be 10, 9, 8, … (includes 10).

Finding Integer Solutions

When asked to list integer values satisfying an inequality, check whether each endpoint is included.
For 3 ≤ x ≤ 6, integers are 3, 4, 5, 6 (both endpoints included).
For 3 ≤ x < 6, integers are 3, 4, 5 (6 not included).
For 3 < x ≤ 6, integers are 4, 5, 6 (3 not included).
For 3 < x < 6, integers are 4, 5 (neither endpoint included).
Remember that zero and negative whole numbers are integers unless stated otherwise.
To satisfy two inequalities, list integers for each and find the intersection (common values).

Representing Inequalities on a Number Line

Use a closed circle (●) for endpoints that are included (≤ or ≥).
Use an open circle (○) for endpoints that are not included (< or >).
Connect circles with a solid line between them for a range.
For a one‑sided inequality (e.g., x > 5), draw an arrow from the circle pointing in the direction of the inequality.
Example: −2 ≤ x < 1 has a closed circle at −2, open circle at 1, and a line between them.
Example: t < 3 has an open circle at 3 and an arrow to the left.

Number line

Solving Simple Inequalities

Inequalities can be solved similarly to equations, but the direction of the inequality sign must be preserved.
Example: Solve y − 2 > 5 → add 2 to both sides → y > 7.
The solution y > 7 means y can be any number greater than 7 (not including 7).
For inequalities involving squares, consider both positive and negative roots (e.g., m² > 9 → m < −3 or m > 3).

Finding the Largest or Smallest Integer

The smallest integer satisfying x > 6.5 is 7 (since 6 is not greater than 6.5).
The largest odd integer satisfying y ≤ 6.5 is 5 (since 7 > 6.5 and 5 is odd).
Always check whether the endpoint is included when determining the extreme integer.

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1.What does the inequality symbol ≤ mean?
Easy
- Aless than or equal to
- Bgreater than or equal to
- Cless than
- Dgreater than
2.Which of the following is a strict inequality?
Easy
- Ax > 5
- Bx ≥ 5
- Cx ≤ 5
- Dx = 5
3.n is an integer. −1 ≤ n < 4. List the possible values of n.
Easy
- A−1, 0, 1, 2, 3
- B−1, 0, 1, 2, 3, 4
- C0, 1, 2, 3
- D−1, 0, 1, 2
4.Write down the inequality shown on the number line: an open circle at 3 and an arrow pointing to the right.
Easy
- Ax > 3
- Bx ≥ 3
- Cx < 3
- Dx ≤ 3
5.m is an integer such that −2 < m ≤ 3. Write down all the possible values of m.
Easy
- A−1, 0, 1, 2, 3
- B−2, −1, 0, 1, 2, 3
- C−1, 0, 1, 2
- D0, 1, 2, 3
6.Show the inequality x < 3 on a number line. Which number line is correct?
Easy
- AOpen circle at 3, arrow left
- BClosed circle at 3, arrow left
- COpen circle at 3, arrow right
- DClosed circle at 3, arrow right
7.n is an integer. Write down all the values of n such that –2 ≤ n < 3.
Easy
- A–2, –1, 0, 1, 2
- B–2, –1, 0, 1, 2, 3
- C–1, 0, 1, 2
- D–2, –1, 0, 1
8.On the number line, represent the inequality y ≤ 1. Which representation is correct?
Easy
- AClosed circle at 1, arrow left
- BOpen circle at 1, arrow left
- CClosed circle at 1, arrow right
- DOpen circle at 1, arrow right

Thêm câu hỏi

9.Circle the inequality shown by the diagram: number line with open circle at –4, closed circle at 5, and a line between them.
Medium
- A–4 < x ≤ 5
- B–4 ≤ x ≤ 5
- C–4 < x < 5
- D–4 ≤ x < 5
10.On the number line, show the inequality –2 ≤ y < 1. Which is correct?
Medium
- AClosed circle at –2, open circle at 1, line between
- BOpen circle at –2, closed circle at 1, line between
- CClosed circle at –2 and 1, line between
- DOpen circle at –2 and 1, line between
11.n is an integer. Write down all the values of n that satisfy –3.4 < n ≤ 2.
Medium
- A–3, –2, –1, 0, 1, 2
- B–3, –2, –1, 0, 1
- C–4, –3, –2, –1, 0, 1, 2
- D–3, –2, –1, 0, 1, 2, 3
12.List all of the integers which satisfy all three inequalities: x ≤ 8, x > 2, x > 3.
Medium
- A4, 5, 6, 7, 8
- B3, 4, 5, 6, 7, 8
- C2, 3, 4, 5, 6, 7, 8
- D4, 5, 6, 7
13.Write down the inequality shown in the diagram: number line with open circle at –2, closed circle at 3, and a line between them.
Medium
- A–2 < x ≤ 3
- B–2 ≤ x ≤ 3
- C–2 < x < 3
- D–2 ≤ x < 3
14.Solve y − 2 > 5.
Easy
- Ay > 7
- By > 3
- Cy < 7
- Dy < 3
15.m² > 9. Circle the possible value of m.
Medium
- A4
- B2
- C2.8
- D3
16.List all of the integers which satisfy all three inequalities: x ≤ 7, 2 < x ≤ 8, 4 ≤ x ≤ 8.5.
Medium
- A4, 5, 6, 7
- B3, 4, 5, 6, 7
- C4, 5, 6, 7, 8
- D5, 6, 7
17.What does the inequality x ≥ 4 mean?
Easy
- Ax is greater than or equal to 4
- Bx is greater than 4
- Cx is less than or equal to 4
- Dx is less than 4
18.Write down the inequality shown on the number line: an open circle at –2 and an arrow pointing to the right.
Easy
- Ax > –2
- Bx ≥ –2
- Cx < –2
- Dx ≤ –2
19.A rectangle has length L cm and width W cm. The length is at most 5 cm and the width is less than 3 cm. Which inequality represents the area A cm² of the rectangle?
Medium
- AA < 15
- BA > 15
- CA ≤ 15
- DA ≥ 15
20.The sum of three consecutive integers is less than 30. What is the largest possible value of the smallest integer?
Hard
- A8
- B9
- C10
- D11
21.A taxi fare is $3.50 plus $1.20 per kilometre. If the fare is at most $15, what inequality represents the distance d km travelled?
Medium
- A3.50 + 1.20d ≤ 15
- B3.50 + 1.20d < 15
- C3.50 + 1.20d ≥ 15
- D3.50 + 1.20d > 15
22.Solve the inequality 3x - 7 ≤ 5x + 1.
Medium
- Ax ≥ -4
- Bx ≤ -4
- Cx ≥ 4
- Dx ≤ 4
23.The diagram shows a triangle with sides 5 cm, 12 cm, and x cm. The triangle is such that x is an integer and the perimeter is less than 30 cm. What is the largest possible value of x?
Hard
- A12
- B13
- C14
- D15
24.A rectangle has length (x+3) cm and width (x-1) cm. Its area is at most 60 cm². Which inequality represents this?
Medium
- A(x+3)(x-1) ≤ 60
- B(x+3)(x-1) < 60
- C(x+3)(x-1) ≥ 60
- D(x+3)(x-1) > 60
25.The inequality 2 < x ≤ 5 is represented on a number line. Which description is correct?
Easy
- AOpen circle at 2, closed circle at 5, line between
- BClosed circle at 2, open circle at 5, line between
- COpen circle at 2, open circle at 5, line between
- DClosed circle at 2, closed circle at 5, line between
26.If x > 4, which of the following is a possible value of x?
Easy
- A3
- B4
- C5
- D3.9
27.The sum of two numbers is at least 10. The larger number is 3 more than the smaller. If the smaller number is y, which inequality represents this?
Medium
- Ay + (y+3) ≥ 10
- By + (y+3) > 10
- Cy + (y+3) ≤ 10
- Dy + (y+3) < 10
28.n is an integer. Which list contains all the integers that satisfy -3 ≤ n < 2?
Easy
- A-3, -2, -1, 0, 1
- B-3, -2, -1, 0, 1, 2
- C-2, -1, 0, 1
- D-2, -1, 0, 1, 2
29.A rectangle has perimeter at most 24 cm. Its length is twice its width. If the width is w cm, which inequality represents the situation?
Medium
- A2(2w + w) ≤ 24
- B2(2w + w) < 24
- C2w + w ≤ 24
- D2w + w < 24
30.Write down the inequality shown on the number line: an open circle at -1 and an arrow pointing to the right.
Easy
- Ax > -1
- Bx ≥ -1
- Cx < -1
- Dx ≤ -1
31.A shop sells pens for $0.50 each and pencils for $0.30 each. A customer buys x pens and y pencils. The total cost is less than $5. Which inequality is correct?
Medium
- A0.5x + 0.3y < 5
- B0.5x + 0.3y ≤ 5
- C0.5x + 0.3y > 5
- D0.5x + 0.3y ≥ 5
32.The inequality -5 < x ≤ 2 is represented on a number line. Which of the following integers satisfy this inequality?
Easy
- A-4, -3, -2, -1, 0, 1, 2
- B-5, -4, -3, -2, -1, 0, 1, 2
- C-4, -3, -2, -1, 0, 1
- D-5, -4, -3, -2, -1, 0, 1
33.A car travels at a speed of v km/h. It covers a distance of at most 200 km in 4 hours. Which inequality represents this?
Medium
- A4v ≤ 200
- B4v < 200
- C4v ≥ 200
- D4v > 200
34.The diagram shows a number line with a closed circle at -2 and an open circle at 3, with a line between them. Which inequality is shown?
Easy
- A-2 ≤ x < 3
- B-2 < x ≤ 3
- C-2 ≤ x ≤ 3
- D-2 < x < 3
35.Write down the inequality shown on the number line: an open circle at 5 and an arrow pointing to the left.
Easy
- Ax < 5
- Bx > 5
- Cx ≤ 5
- Dx ≥ 5
36.n is an integer. Write down all the possible values of n such that –3 < n ≤ 1.
Easy
- A–2, –1, 0, 1
- B–3, –2, –1, 0, 1
- C–2, –1, 0
- D–3, –2, –1, 0
37.On the number line, represent the inequality y ≥ 2. Which representation is correct?
Easy
- AA closed circle at 2 with an arrow to the right
- BAn open circle at 2 with an arrow to the right
- CA closed circle at 2 with an arrow to the left
- DAn open circle at 2 with an arrow to the left
38.Solve the inequality 4x + 3 > 15.
Medium
- Ax > 3
- Bx > 4.5
- Cx < 3
- Dx < 4.5
39.The inequality x > 5 means x is greater than or equal to 5.
Easy
True or false?
40.The inequality x ≥ 4 means x is greater than 4.
Easy
True or false?
41.Which number line correctly represents the inequality x ≤ 2?
Easy
- AA closed circle at 2 with an arrow pointing to the left.
- BAn open circle at 2 with an arrow pointing to the left.
- CA closed circle at 2 with an arrow pointing to the right.
- DAn open circle at 2 with an arrow pointing to the right.
42.On a number line, an open circle at an endpoint means that endpoint is included in the inequality.
Easy
True or false?
43.Match each inequality symbol with its meaning.
Easy
- >
- <
- ≥
- ≤
- less than
- greater than
- less than or equal to
- greater than or equal to
44.Arrange the following inequality symbols in order from 'strictly less than' to 'strictly greater than'.
Easy
- <
- ≤
- >
- ≥
45.Complete the sentence: The inequality symbol ≤ means ________.
Easy
The inequality symbol ≤ means ____.
46.Find the smallest integer that satisfies x > 6.5.
Easy
47.Find the smallest integer that satisfies x > 6.5.
Easy
- A8
- B7
- C9
- D6

Ghi chú ôn tập

## Inequalities ### Interpreting Inequalities - **Inequality**: compares values using >, <, ≥, ≤. - **Strict inequalities**: > and < (endpoint not included). - **Non-strict**: ≥ and ≤ (endpoint included). #### Finding integer values - List whole numbers between endpoints, checking inclusion. - Example: −4 ≤ x < 2 → x = −4, −3, −2, −1, 0, 1. - For two inequalities, list integers satisfying both. - If no integer specified, do not assume integer. ### Representing on a Number Line - **Open circle** (○) for < or > (not included). - **Closed circle** (●) for ≤ or ≥ (included). - Draw a line connecting circles, or an arrow for one endpoint. - Example: −2 ≤ x < 1 → closed at −2, open at 1, line between. - Example: t < 3 → open at 3, arrow left. ### Key Points - Integers include zero and negatives. - For smallest/largest integer: e.g., smallest integer > 6.5 is 7.

Inequality: A statement that one value is greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) another.
Strict inequality: An inequality using > or <; the endpoint is not included.
Integer: A whole number (positive, negative, or zero).
Number line: A visual representation of numbers; used to show inequalities with open/closed circles and lines or arrows.
Open circle: A circle drawn above a number on a number line to indicate that the endpoint is not included (used with < or >).
Closed circle: A filled circle drawn above a number on a number line to indicate that the endpoint is included (used with ≤ or ≥).

Thẻ ghi nhớ

What does the inequality x > 5 mean?

x is greater than 5. x could be 6, 7, 8, 9, ...

What does the symbol ⩾ mean?

Greater than or equal to.

What does the symbol ⩽ mean?

Less than or equal to.

What are strict inequalities?

Inequalities that do not include being equal, i.e., > and <.

List the integer values of n that satisfy −1 ⩽ n < 4.

n = -1, 0, 1, 2, 3

List the integer values of y that satisfy −5 < y ⩽ 0.

y = -4, -3, -2, -1, 0

List the integer values of m that satisfy −2 < m ⩽ 3.

m = -1, 0, 1, 2, 3

How do you represent an inequality on a number line?

Draw circles above the end points: open circle for < or >, solid circle for ⩽ or ⩾. Connect them with a horizontal line, or use an arrow for one end point.

Show the inequality x < 3 on a number line.

An open circle at 3 with a horizontal arrow pointing to the left.

Represent the inequality y ⩽ 1 on a number line.

A solid circle at 1 with a horizontal arrow pointing to the left.

Which inequality is shown by a diagram with a solid circle at -4 and an open circle at 5, connected by a line?

−4 ⩽ x < 5

Show the inequality −2 ⩽ y < 1 on a number line.

Solid circle at -2, open circle at 1, with a line connecting them.

List all integer values of n that satisfy −3.4 < n ⩽ 2.

n = -3, -2, -1, 0, 1, 2

List all integers that satisfy x ≤ 8, x > 2, and x > 3.

x = 4, 5, 6, 7, 8

What is the largest odd integer that satisfies y ≤ 6.5?

Write down the inequality shown in a diagram with an open circle at -2 and a solid circle at 3, connected by a line.

−2 < n ⩽ 3

Solve y − 2 > 5.

y > 7

Represent the inequality −2 < n ⩽ 3 on a number line.

Open circle at -2, solid circle at 3, with a line connecting them.

Given m² > 9, circle the possible values of m from: 2, 7, 8, 2.8, 3, −7.

7, 8, and −7 (m² > 9 means m < −3 or m > 3).

List all integers that satisfy x ≤ 7, 2 < x ≤ 8, and 4 ≤ x ≤ 8.5.

x = 4, 5, 6, 7

What is the smallest integer that satisfies x > 6.5?

List the integer values of x that satisfy −4 ≤ x < 2.

x = -4, -3, -2, -1, 0, 1

Represent the inequality t < 3 on a number line.

Open circle at 3 with a horizontal arrow pointing to the left.

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