Inequalities

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Apuntes de la lección

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.

Diapositivas

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Preguntas de práctica

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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

Más preguntas

9.Which of the following inequalities are satisfied by x = 2? (select all that apply)
Medium
- Ax > 1
- Bx < 2
- Cx ≥ 2
- Dx ≤ 2
- Ex > 3
10.On a number line, which type of circle is used to show that the endpoint is NOT included in the solution of an inequality?
Easy
- AAn open circle
- BA closed circle
- CA shaded circle
- DA square
11.The inequality x ≥ 3 is shown on a number line. Which statements about the diagram are true? (select all that apply)
Medium
- AThe circle at 3 is closed
- BThe circle at 3 is open
- CThe arrow points to the right of 3
- DThe arrow points to the left of 3
- EThe value 3 itself is included in the solution set
12.Match each inequality on the left with the description of its number-line diagram on the right.
Easy
- x > 4
- x ≤ 4
- x < 4
- x ≥ 4
- Open circle at 4, arrow to the right
- Closed circle at 4, arrow to the left
- Open circle at 4, arrow to the left
- Closed circle at 4, arrow to the right
13.Starting with the smallest, order the integer values that satisfy the inequality −2 ≤ x < 3.
Medium
- -2
- -1
- 0
- 1
- 2
14.Which of the following inequalities are satisfied by x = −3? (select all that apply)
Medium
- Ax > -4
- Bx ≤ -3
- Cx ≥ -2
- Dx < -3
- Ex > -2
15.The inequality a ≤ b means that a is at most b.
Easy
True or false?
16.Solve the inequality 3x < 12.
Medium
- Ax < 4
- Bx > 4
- Cx < 36
- Dx > 36
17.Solve the inequality 5 − 2x > 11.
Medium
- Ax < -3
- Bx > -3
- Cx < 3
- Dx > 3
18.Which inequality is represented on the number line?
Medium
- A-1 ≤ x < 4
- B-1 < x ≤ 4
- C-1 ≤ x ≤ 4
- D-1 < x < 4
19.Which of these values satisfy the inequality shown? (select all that apply)
Medium
- A-1
- B0
- C3.5
- D4
- E5
20.Solve the inequality 4x ≤ 20.
Easy
- Ax ≤ 5
- Bx ≥ 5
- Cx ≤ 80
- Dx ≥ 80
21.The inequality 6 > 4 is equivalent to 4 < 6.
Easy
True or false?
22.Solve the inequality 7 − x ≥ 3.
Medium
- Ax ≤ 4
- Bx ≥ 4
- Cx ≤ 10
- Dx ≥ 10
23.Which of the following integers satisfy the inequality 2x < 9? (select all that apply)
Medium
- A4
- B5
- C0
- D−1
- E6
24.Match each inequality to the smallest integer that satisfies it.
Medium
- x ≥ 5
- x > 5
- x ≤ 5
- x < 5
- 5
- 6
- 4
25.Solve the inequality −3x < 12.
Medium
- Ax > −4
- Bx < −4
- Cx > 4
- Dx < 4
26.What is the first step to solve this inequality?
Medium
- ASubtract 2 from both sides
- BAdd 2 to both sides
- CDivide both sides by 5
- DMultiply both sides by 5
27.A taxi charges a fixed fee of £3 plus £2 per mile. Priya has £23. If she travels m miles, which inequality must hold?
Medium
- A3 + 2m ≤ 23
- B3 + 2m ≥ 23
- C2 + 3m ≤ 23
- D3m + 2 ≤ 23
28.A rectangle has width w cm and length 5 cm more than its width. Its perimeter must be at least 34 cm. Which inequality models this?
Hard
- A2w + 2(w + 5) ≥ 34
- B2w + 2(w + 5) ≤ 34
- Cw + (w + 5) ≥ 34
- D2w + 2(w − 5) ≥ 34
29.Solve the inequality 5 − 3x ≥ 20.
Hard
- Ax ≤ −5
- Bx ≥ −5
- Cx ≤ 5
- Dx ≥ 5
30.Which of the following integers satisfy the inequality 3x − 4 < 8? (select all that apply)
Medium
- A0
- B3
- C4
- D−2
- E5
31.If 4 − x > 1, then x < 3.
Easy
True or false?
32.Match each inequality to the largest integer value of x that satisfies it.
Medium
- x + 4 < 9
- 2x ≤ 11
- 3x − 1 < 14
- 10 − x ≥ 4
- 4
- 5
- 6
33.Order these inequalities from the smallest largest-satisfying integer value of x to the largest.
Medium
- x + 2 < 7
- 2x ≤ 14
- 3x − 4 < 8
- x − 1 > 5
34.Which symbol means 'is greater than'?
Easy
- A<
- B>
- C≤
- D≠
35.Which symbol means 'is at most'?
Easy
- A<
- B>
- C≤
- D≥
36.Which of these is a strict inequality?
Easy
- Ax ≤ 4
- Bx ≥ 4
- Cx < 4
- Dx = 4
37.Which of the following are true statements? (select all that apply)
Easy
- A5 > 3
- B2 < 2
- C7 ≤ 7
- D9 ≥ 10
- E−1 > −4
38.The statement 3 < 8 is equivalent to 8 > 3.
Easy
True or false?
39.The inequality x ≤ 5 is satisfied by x = 5.
Easy
True or false?
40.Match each inequality symbol with its meaning in words.
Medium
- <
- >
- ≤
- ≥
- is at least
- is less than
- is at most
- is greater than
41.Solve the inequality x + 5 < 12. Give the largest integer value of x that satisfies it.
Medium
42.Complete the statement about solving inequalities.
Easy
When you multiply or divide both sides of an inequality by a negative number, you must ____ the inequality sign.
43.Solve the inequality 2x + 1 ≤ 9. What is the largest integer value of x that satisfies it?
Medium
44.Complete the sentence about solving an inequality.
Medium
When you multiply or divide both sides of an inequality by a negative number, you must ____ the inequality sign.
45.How many integer values satisfy the inequality shown?
Medium
46.Solve the inequality x − 7 > 2. Give the smallest integer value of x that satisfies it.
Medium
47.Complete the statement.
Easy
When both sides of an inequality are multiplied by a negative number, the inequality sign must be ____.
48.Solve the inequality 3x + 4 ≤ 19. Give the largest integer value of x that satisfies it.
Medium
49.Write the inequality that means 'x is at most 12'.
Medium
50.Solve the inequality. Give the largest integer value of x that satisfies it.
Medium
51.Solve the inequality 2x + 5 < 17. Give the largest integer value of x that satisfies it.
Medium
52.What is the name of the inequality symbol ≤?
Medium
53.Explain why the inequality sign must be reversed when multiplying or dividing both sides of an inequality by a negative number.
Hard
54.Explain why the inequality sign reverses when you divide both sides of an inequality by a negative number.
Hard
55.Explain how to show the inequality x ≥ 2 on a number line, including the type of circle used.
Hard
56.A bag contains n counters. There are 4 more red counters than blue counters, and the total number of counters is fewer than 30. Write and solve an inequality to find the greatest possible number of blue counters.
Hard
57.Explain why the inequality sign must be reversed when both sides of an inequality are multiplied by a negative number.
Medium
58.Solve the inequality x + 5 < 12. Give the largest integer value of x that satisfies it.
Medium
- A8
- B6
- C5
- D7
59.Solve the inequality 2x + 1 ≤ 9. What is the largest integer value of x that satisfies it?
Medium
- A4
- B5
- C3
- D6
60.How many integer values satisfy the inequality shown?
Medium
- A7
- B6
- C4
- D5
61.Solve the inequality x − 7 > 2. Give the smallest integer value of x that satisfies it.
Medium
- A11
- B10
- C12
- D9
62.Solve the inequality 3x + 4 ≤ 19. Give the largest integer value of x that satisfies it.
Medium
- A5
- B4
- C7
- D6
63.Solve the inequality. Give the largest integer value of x that satisfies it.
Medium
- A1
- B2
- C3
- D4
64.Solve the inequality 2x + 5 < 17. Give the largest integer value of x that satisfies it.
Medium
- A4
- B7
- C6
- D5

Apuntes de repaso

## 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.

Tarjetas

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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