Linear Equations And Inequalities

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Solving Linear Equations

A linear equation has the form ax + b = c where the highest power of x is 1.
To solve, use inverse operations: add/subtract to undo addition/subtraction, multiply/divide to undo multiplication/division.
Any operation performed on one side must be performed on the other side.
When solving, it is often easier to remove the smallest x term to avoid negatives.
If the equation contains brackets, expand them first (e.g., 2(x-3)=10 → 2x-6=10).
If the equation contains fractions, multiply both sides by the lowest common denominator.
If the unknown is in the denominator, multiply both sides by that denominator.
Always check your solution by substituting back into the original equation.

Solving Linear Equations with x on Both Sides

Collect x terms on one side by adding or subtracting the smaller x term from both sides.
Example: 4x - 7 = 11 + x → subtract x: 3x - 7 = 11 → add 7: 3x = 18 → divide by 3: x = 6.
If the smaller x term is negative, add its positive value to both sides (e.g., 4 - 5x = 6x - 29 → add 5x: 4 = 11x - 29).
After collecting x terms, solve using inverse operations as usual.

Solving Linear Inequalities

An inequality compares values using >, <, ≥, or ≤.
Strict inequalities (<, >) do not include the endpoint; non-strict (≤, ≥) do.
Solve inequalities exactly like equations, but keep the inequality sign throughout.
Critical rule: When multiplying or dividing by a negative number, reverse the inequality sign (e.g., -x > 2 → x < -2).
Never multiply or divide by a variable (x) because its sign is unknown.
For double inequalities (e.g., a < 2x < b), apply the same operation to all three parts, or split into two separate inequalities.

Integer Solutions to Inequalities

When asked for integer values satisfying an inequality, list all whole numbers within the range.
Pay attention to whether endpoints are included: ≤ or ≥ means included; < or > means excluded.
Example: -3 ≤ x < 1 gives integers x = -3, -2, -1, 0.
If two inequalities are given, find the intersection of their solution sets.
If only one endpoint is given, there are infinitely many integers (e.g., x > 2 gives 3, 4, 5, ...).
Remember that 0 and negative numbers are integers unless specified otherwise.

Representing Inequalities on a Number Line

Use an open circle for strict inequalities (< or >) to show the endpoint is not included.
Use a closed (filled) circle for non-strict inequalities (≤ or ≥) to show inclusion.
Draw a horizontal line or arrow connecting the circles to indicate the range.
For inequalities with only one endpoint, draw an arrow extending to the left (for < or ≤) or right (for > or ≥).

Number line

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1.Solve the equation: 3w − 7 = 32.
Easy
- Aw = 13
- Bw = 11
- Cw = 39
- Dw = 8.33
2.Solve the inequality: 7m − 2 ≥ 19.
Easy
- Am ≥ 3
- Bm ≥ 21
- Cm ≥ 2.43
- Dm ≥ 17
3.Write down the integer values of x that satisfy the inequality −3 ≤ x < 1.
Easy
- A−3, −2, −1, 0
- B−3, −2, −1, 0, 1
- C−2, −1, 0
- D−3, −2, −1
4.Solve the equation: 6x − 3 = −12.
Easy
- Ax = −1.5
- Bx = −2.5
- Cx = 2.5
- Dx = −15
5.Solve the equation: 1 − x/3 = 5.
Easy
- Ax = −12
- Bx = 12
- Cx = −2
- Dx = 2
6.Solve the inequality: n + 7 < 5n − 8.
Medium
- An > 3.75
- Bn < 3.75
- Cn > −0.25
- Dn < −0.25
7.Solve the equation: 5x − 17 = 7x + 3.
Medium
- Ax = −10
- Bx = 10
- Cx = −7
- Dx = 7
8.Solve the equation: 4(5 − 3x) = 23.
Medium
- Ax = −0.25
- Bx = 0.25
- Cx = −3.58
- Dx = 3.58

Thêm câu hỏi

9.Find the integer values of n that satisfy the inequality 18 − 2n < 6n ≤ 30 + n.
Hard
- An = 3, 4, 5, 6
- Bn = 3, 4, 5
- Cn = 4, 5, 6
- Dn = 2, 3, 4, 5
10.Solve the equation: 9f + 11 = 3f + 23.
Easy
- Af = 2
- Bf = 3
- Cf = 4
- Df = 1
11.Find the integer values of x that satisfy the inequality 2 < 2x ≤ 10.
Medium
- Ax = 2, 3, 4, 5
- Bx = 1, 2, 3, 4, 5
- Cx = 2, 3, 4, 5, 6
- Dx = 1, 2, 3, 4
12.Solve the equation: 6 − 2x = 3x.
Easy
- Ax = 1.2
- Bx = 1.5
- Cx = 2
- Dx = 0.6
13.Solve the equation: 2/x = 6/(2 − x).
Medium
- Ax = 0.5
- Bx = 1
- Cx = 2
- Dx = 0.25
14.A rectangle has length (x+5) cm and width (x−2) cm. Its perimeter is 34 cm. Find the value of x.
Medium
- Ax = 7
- Bx = 8
- Cx = 9
- Dx = 10
15.The sum of three consecutive integers is 42. Find the smallest integer.
Medium
- A13
- B14
- C15
- D12
16.Solve the inequality 3(x−2) < 7(x+2).
Medium
- Ax > −5
- Bx < −5
- Cx > 5
- Dx < 5
17.Solve the equation 15/x = −3.
Easy
- Ax = −5
- Bx = 5
- Cx = −45
- Dx = 45
18.Find the integer values of n that satisfy the inequality −7 < 4n ≤ 8.
Medium
- An = −1, 0, 1, 2
- Bn = −1, 0, 1
- Cn = 0, 1, 2
- Dn = −2, −1, 0, 1, 2
19.Solve the inequality 3m + 12 ≤ 8m − 5.
Medium
- Am ≥ 3.4
- Bm ≤ 3.4
- Cm ≥ 17/5
- Dm ≤ 17/5
20.Solve the equation 3(x−4) + (x+2)/5 = 6.
Medium
- Ax=5.5
- Bx=6
- Cx=7
- Dx=8
21.A triangle has sides of lengths x cm, (x+3) cm, and (2x−1) cm. Its perimeter is 22 cm. Find x.
Medium
- Ax = 5
- Bx = 4
- Cx = 6
- Dx = 7
22.Solve the inequality 4x + 5 ≤ 17.
Easy
- Ax ≤ 3
- Bx ≥ 3
- Cx ≤ 4
- Dx ≥ 4
23.The result of a survey shows that the number of students, n, satisfies 15 ≤ 4n < 28. Find the possible integer values of n.
Medium
- A4, 5, 6
- B3, 4, 5, 6
- C4, 5, 6, 7
- D3, 4, 5
24.Solve the equation 1 − p/3 = 4.
Medium
- Ap = −9
- Bp = 9
- Cp = −15
- Dp = 15
25.Solve the inequality 18 − 2n < 6n ≤ 30 + n.
Hard
- A9/4 < n ≤ 6
- B9/4 ≤ n ≤ 6
- Cn > 9/4
- Dn ≤ 6
26.A linear equation can contain terms like x².
Easy
True or false?
27.What is the first step to solve the equation 3x + 5 = 14?
Easy
- ASubtract 5 from both sides
- BAdd 5 to both sides
- CMultiply both sides by 3
- DDivide both sides by 3
28.Which inequality represents 'x is greater than or equal to 4'?
Easy
- Ax > 4
- Bx < 4
- Cx ≥ 4
- Dx ≤ 4
29.What is the inverse operation of multiplying by 5?
Easy
- AAdding 5
- BSubtracting 5
- CDividing by 5
- DMultiplying by 5 again
30.When solving an inequality, multiplying both sides by a negative number reverses the inequality sign.
Easy
True or false?
31.Match each inequality with its representation on a number line (open or closed circle).
Medium
- x > 3
- x ≥ 3
- x < 3
- x ≤ 3
- Open circle at 3, arrow right
- Closed circle at 3, arrow right
- Open circle at 3, arrow left
- Closed circle at 3, arrow left
32.Which of the following are integers that satisfy the inequality −3 < x ≤ 2? (Select all that apply)
Medium
- A-3
- B-2
- C-1
- D0
- E1
- F2
33.Arrange the steps in order to solve the equation 2x + 3 = 11.
Easy
- Subtract 3 from both sides
- Divide both sides by 2
- Write the solution x = 4
34.Match each inequality with its correct representation on a number line.
Medium
- x > 3
- x ≤ -2
- -1 < x ≤ 4
- Open circle at 3, arrow to the right
- Closed circle at -2, arrow to the left
- Open circle at -1, closed circle at 4, line between
35.Arrange the steps in the correct order to solve the equation 3x - 7 = 11.
Medium
- Add 7 to both sides
- Divide both sides by 3
- Simplify to 3x = 18
- Simplify to x = 6
36.Solve the equation: x + 7 = 12.
Easy
37.Complete the sentence.
Easy
The smallest integer that satisfies x > 6.5 is ____.
38.Solve the inequality: 5x − 3 ≤ 12.
Medium
39.Solve the inequality 3(x - 4) > 2(2x - 5). Write your answer as x > a or x < a, giving a as a simplified fraction.
Hard
40.Solve the equation: x + 7 = 12.
Easy
- A6
- B7
- C5
- D4
41.Solve the inequality: 5x − 3 ≤ 12.
Easy
- A3
- B2
- C4
- D5
42.Solve the inequality 3(x - 4) > 2(2x - 5). Write your answer as x > a or x < a, giving a as a simplified fraction.
Medium
- A0
- B-1
- C-2
- D-3

Ghi chú ôn tập

## Linear Equations & Inequalities ### Solving Linear Equations - **Form:** ax + b = c (x power 1). - **Inverse operations:** add/subtract, multiply/divide (do same to both sides). - **Order:** any order works; easier to undo addition/subtraction first. - **Brackets:** expand first, e.g., 2(x-3)=10 → 2x-6=10. - **Fractions:** multiply by lowest common denominator, e.g., x/5 + 4 = 9/2 → multiply by 10 → 2x+40=45. - **Unknown on denominator:** multiply both sides by denominator, e.g., 4/(x-2)=3 → 4=3(x-2). - **x on both sides:** collect x terms on one side (remove smaller x term to avoid negatives). - **Check:** substitute answer back into original equation. ### Solving Linear Inequalities - **Inequality signs:** >, <, ≥, ≤. Strict: >, < (not equal). - **Solving:** same as equations, but keep inequality sign throughout. - **Multiplying/dividing by negative:** **flip the inequality sign** (e.g., -3x < 6 → x > -2). - **Never multiply/divide by a variable** (unknown sign). - **Double inequalities:** apply same operation to all three parts, e.g., a < 2x < b → a/2 < x < b/2. - **Integer solutions:** list whole numbers satisfying the inequality; note inclusion of endpoints (≤ or ≥ includes, < or > excludes). - **Number line:** open circle for < or >; closed circle for ≤ or ≥; arrow for one-sided. ### Key Points - **Integer values:** include zero and negatives unless specified "positive integers". - **Smallest/largest integer:** e.g., smallest integer satisfying x > 6.5 is 7. - **Do not assume x is integer** unless stated.

Linear equation: An equation of the form ax + b = c, where a, b, c are numbers and x is a variable with highest power 1.
Inverse operations: Operations that undo each other: addition/subtraction, multiplication/division.
Inequality: A statement that one quantity is greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) another.
Strict inequality: An inequality using > or <, where 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 arrows.
Double inequality: An inequality with three parts, e.g., a < x < b, solved by applying operations to all three sides.
Flip the sign: When multiplying or dividing an inequality by a negative number, reverse the inequality direction.

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