Grade 9 Exam > Grade 9 Notes > Integrated Math 1 > Cheatsheet: Solving Equations & Inequalities
Cheatsheet: Solving Equations & Inequalities
1. One-Step Equations
1.1 Basic Operations
| Operation Type | Method & Example |
|---|---|
| Addition | Subtract the same number from both sides: x + 5 = 12 → x = 7 |
| Subtraction | Add the same number to both sides: x - 3 = 9 → x = 12 |
| Multiplication | Divide both sides by the coefficient: 4x = 20 → x = 5 |
| Division | Multiply both sides by the divisor: x/3 = 6 → x = 18 |
1.2 Key Principles
- Inverse operations undo each other: addition ↔ subtraction, multiplication ↔ division
- Perform the same operation on both sides to maintain equality
- Check solution by substituting back into original equation
2. Two-Step Equations
2.1 Solution Process
| Step | Action |
|---|---|
| 1. Eliminate constant | Use addition or subtraction to move constant term to other side |
| 2. Eliminate coefficient | Use multiplication or division to isolate variable |
2.2 Example Pattern
- 3x + 7 = 22
- Step 1: 3x = 15 (subtract 7 from both sides)
- Step 2: x = 5 (divide both sides by 3)
3. Multi-Step Equations
3.1 Order of Operations for Solving
- Distribute if needed
- Combine like terms on each side
- Move variable terms to one side
- Move constant terms to other side
- Divide or multiply to isolate variable
3.2 Distributive Property
| Form | Example |
|---|---|
| a(b + c) = ab + ac | 2(x + 3) = 2x + 6 |
| a(b - c) = ab - ac | -3(x - 4) = -3x + 12 |
3.3 Combining Like Terms
- Like terms have the same variable with same exponent: 3x and 5x are like terms
- Add or subtract coefficients: 3x + 5x = 8x
- Constants are like terms with each other: 7 + 4 = 11
4. Equations with Variables on Both Sides
4.1 Solution Strategy
- Simplify each side separately
- Move all variable terms to one side
- Move all constants to other side
- Solve for variable
4.2 Example
- 5x + 3 = 2x + 12
- 3x + 3 = 12 (subtract 2x from both sides)
- 3x = 9 (subtract 3 from both sides)
- x = 3 (divide both sides by 3)
4.3 Special Cases
| Case | Meaning |
|---|---|
| Identity (0 = 0 or 5 = 5) | Infinitely many solutions; all real numbers work |
| Contradiction (0 = 5 or 3 = 7) | No solution; equation is impossible |
5. Solving Inequalities
5.1 Inequality Symbols
| Symbol | Meaning |
|---|---|
| < | Less than |
| > | Greater than |
| ≤ | Less than or equal to |
| ≥ | Greater than or equal to |
5.2 Solving Rules
- Use same steps as solving equations
- Addition/subtraction: inequality sign stays the same
- Multiplication/division by positive number: inequality sign stays the same
- Multiplication/division by negative number: reverse the inequality sign
5.3 Critical Rule
| Operation | Effect on Inequality |
|---|---|
| Multiply or divide by negative | FLIP the inequality sign: -2x < 6 → x > -3 |
| Multiply or divide by positive | Keep inequality sign same: 2x < 6 → x < 3 |
6. Graphing Solutions on Number Line
6.1 Inequality Notation
| Inequality | Graph Symbol |
|---|---|
| x < a or x > a | Open circle at a (not included) |
| x ≤ a or x ≥ a | Closed/filled circle at a (included) |
| x < a or x ≤ a | Shade to the left of a |
| x > a or x ≥ a | Shade to the right of a |
6.2 Interval Notation
| Inequality | Interval Notation |
|---|---|
| x < 3 | (-∞, 3) |
| x ≤ 3 | (-∞, 3] |
| x > 3 | (3, ∞) |
| x ≥ 3 | [3, ∞) |
- Parentheses ( ) mean not included (open circle)
- Brackets [ ] mean included (closed circle)
- ∞ and -∞ always use parentheses
7. Compound Inequalities
7.1 Types
| Type | Meaning |
|---|---|
| AND (conjunction) | Both conditions must be true; solution is intersection |
| OR (disjunction) | At least one condition must be true; solution is union |
7.2 AND Inequalities
- Form: a < x < b or x > a AND x < b
- Example: -2 < x < 5 means x is between -2 and 5
- Interval notation: (-2, 5)
- Graph: shade region between both values
7.3 OR Inequalities
- Form: x < a OR x > b
- Example: x < -2 OR x > 5
- Interval notation: (-∞, -2) ∪ (5, ∞)
- Graph: shade two separate regions
8. Absolute Value Equations
8.1 Definition
- |x| = distance from zero on number line
- |x| is always non-negative: |5| = 5 and |-5| = 5
8.2 Solving Method
| Equation Form | Solution Method |
|---|---|
| |x| = a (where a ≥ 0) | Two solutions: x = a or x = -a |
| |x| = a (where a < 0) | No solution (absolute value cannot be negative) |
8.3 Steps for |ax + b| = c
- Isolate absolute value expression
- Set up two equations: ax + b = c and ax + b = -c
- Solve each equation separately
- Check both solutions in original equation
8.4 Example
- |2x - 3| = 7
- Case 1: 2x - 3 = 7 → 2x = 10 → x = 5
- Case 2: 2x - 3 = -7 → 2x = -4 → x = -2
- Solutions: x = 5 or x = -2
9. Absolute Value Inequalities
9.1 Less Than Inequalities
| Form | Solution |
|---|---|
| |x| < a | -a < x < a (AND compound inequality) |
| |x| ≤ a | -a ≤ x ≤ a (AND compound inequality) |
9.2 Greater Than Inequalities
| Form | Solution |
|---|---|
| |x| > a | x < -a OR x > a (OR compound inequality) |
| |x| ≥ a | x ≤ -a OR x ≥ a (OR compound inequality) |
9.3 Examples
- |x| < 3 → -3 < x < 3
- |x| > 3 → x < -3 OR x > 3
- |x + 2| ≤ 5 → -5 ≤ x + 2 ≤ 5 → -7 ≤ x ≤ 3
- |x - 4| > 2 → x - 4 < -2 OR x - 4 > 2 → x < 2 OR x > 6
10. Literal Equations
10.1 Definition
- Equations with multiple variables
- Solve for one variable in terms of others
- Formulas are examples of literal equations
10.2 Solution Strategy
- Treat other variables as constants
- Use same solving steps as regular equations
- Isolate the desired variable
10.3 Common Formulas
| Formula | Solved for Different Variable |
|---|---|
| A = lw | l = A/w or w = A/l |
| P = 2l + 2w | l = (P - 2w)/2 or w = (P - 2l)/2 |
| d = rt | r = d/t or t = d/r |
| C = 2πr | r = C/(2π) |
| A = ½bh | b = 2A/h or h = 2A/b |
11. Common Mistakes to Avoid
11.1 Equation Errors
- Forgetting to distribute negative sign: -(x - 3) = -x + 3, not -x - 3
- Not performing operation on both sides
- Incorrectly combining unlike terms: 2x + 3 cannot be simplified further
- Division by zero is undefined
11.2 Inequality Errors
- Forgetting to flip sign when multiplying/dividing by negative
- Using wrong circle type on graph (open vs. closed)
- Confusing AND with OR in compound inequalities
11.3 Absolute Value Errors
- Forgetting the negative case when solving equations
- Confusing less than (AND) with greater than (OR) for inequalities
- Not checking if solution makes absolute value negative
12. Checking Solutions
12.1 Verification Steps
- Substitute solution back into original equation/inequality
- Simplify both sides
- Confirm both sides are equal (for equations) or inequality is true
- For inequalities, test with a value in solution set
12.2 Why Check
- Catches arithmetic errors
- Identifies extraneous solutions
- Confirms inequality direction is correct
- Verifies absolute value solutions are valid
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Apr 26, 2026 Last updated
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