Linear Equations And Inequalities

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Lesson notes

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

Slides

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Practice questions

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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

Revision notes

## 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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