Inequalities

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课程笔记

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.

幻灯片

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练习题

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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

复习笔记

## 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 ≥).

闪卡

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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