Mathematics · Topic 1.2
Equations, Inequalities & Functions
Build on your knowledge of expressions to solve multi-step equations, work with linear inequalities, and explore the concept of functions on the coordinate plane.
What You'll Learn
- Solve one- and two-step linear equations using inverse operations
- Solve linear inequalities and represent solutions on a number line
- Understand why the inequality sign reverses when multiplying/dividing by a negative
- Define a function and distinguish it from a non-function
- Plot ordered pairs and identify quadrants on the coordinate plane
- Graph simple linear functions and interpret slope and y-intercept
IB Assessment Focus
Criterion A: Solve equations and inequalities accurately in familiar and unfamiliar contexts.
Criterion B: Investigate patterns in input-output tables and describe them as function rules.
Criterion C: Communicate algebraic reasoning clearly using correct notation.
Criterion D: Model real-world problems with equations, inequalities, or functions.
Key Vocabulary
| Term | Definition |
|---|---|
| Equation | A mathematical statement that two expressions are equal, containing an unknown variable |
| Inverse operation | The operation that undoes another (addition ↔ subtraction; multiplication ↔ division) |
| Linear inequality | A statement comparing two expressions using <, >, ≤, or ≥ |
| Solution set | All values of x that make an equation or inequality true |
| Function | A rule that assigns exactly ONE output to each input |
| Input (x) | The independent variable; the value you put in |
| Output (y) | The dependent variable; the value the function produces |
| Coordinate plane | The grid formed by a horizontal x-axis and vertical y-axis |
| Ordered pair | A pair of values (x, y) that locates a point on the coordinate plane |
| Quadrant | One of four regions of the coordinate plane (I, II, III, IV) |
Mathematics · Topic 1.2
Solving Linear Equations
An equation states that two expressions are equal. Solving means finding the value of the unknown that makes the equation true.
The Balance Method
Think of an equation as a balance scale. Whatever you do to one side, you must do to the other to keep it balanced.
Strategy
Use inverse operations to isolate the variable. Undo addition with subtraction, undo multiplication with division — always doing the same to both sides.
One-Step Equations
Example: x + 7 = 15
Subtract 7 from both sides: x + 7 − 7 = 15 − 7
x = 8
Example: 3x = −18
Divide both sides by 3: 3x ÷ 3 = −18 ÷ 3
x = −6
Two-Step Equations
Undo operations in reverse order: deal with addition/subtraction first, then multiplication/division.
Example: 2x + 5 = 17
Subtract 5 from both sides: 2x = 12
Divide both sides by 2: x = 6
Check: 2(6) + 5 = 12 + 5 = 17 ✓
Equations with Variables on Both Sides
Example: 5x − 3 = 2x + 9
Subtract 2x from both sides: 3x − 3 = 9
Add 3 to both sides: 3x = 12
Divide both sides by 3: x = 4
Check: 5(4) − 3 = 17 and 2(4) + 9 = 17 ✓
Always Check: Substitute your answer back into the original equation to verify. This catches sign errors and arithmetic mistakes.
Mathematics · Topic 1.2
Linear Inequalities
An inequality is like an equation but uses comparison symbols instead of an equals sign. The solution is a range of values, not a single number.
Inequality Symbols
| Symbol | Meaning | Number Line |
|---|---|---|
| < | Less than | Open circle ○, shade left |
| > | Greater than | Open circle ○, shade right |
| ≤ | Less than or equal to | Closed circle ●, shade left |
| ≥ | Greater than or equal to | Closed circle ●, shade right |
Solving Inequalities
Solve exactly like an equation — with one critical exception.
The Sign-Flip Rule
When you multiply or divide both sides by a negative number, you must REVERSE the inequality sign.
Example: −3x + 9 ≤ 21
Subtract 9 from both sides: −3x ≤ 12
Divide by −3 (flip the sign!): x ≥ −4
Number line: closed circle at −4, shade to the right
Example: 4x + 2 > 18
Subtract 2 from both sides: 4x > 16
Divide by 4 (positive → no flip): x > 4
Number line: open circle at 4, shade to the right
Most Common Error: Forgetting to flip the inequality sign when dividing by a negative. Example: −2x < 6 → x > −3 (not x < −3). This is the number-one algebra mistake at Grade 7.
Why Does the Sign Flip?
Consider the true statement 2 < 5. Multiply both sides by −1: −2 and −5. But −2 > −5, so the direction must reverse. Multiplying by a negative reverses the order of numbers on the number line.
Mathematics · Topic 1.2
Functions & the Coordinate Plane
A function is a special relationship where each input produces exactly one output. The coordinate plane lets us visualise this relationship.
What Is a Function?
- A function assigns exactly one output to each input
- If one input gives two different outputs, it is NOT a function
- Notation: f(x) = 3x − 1 means "the function f takes input x, multiplies by 3, then subtracts 1"
Function Machine Example: f(x) = 2x + 3
| Input (x) | Rule: 2x + 3 | Output f(x) |
|---|---|---|
| −2 | 2(−2) + 3 | −1 |
| 0 | 2(0) + 3 | 3 |
| 1 | 2(1) + 3 | 5 |
| 4 | 2(4) + 3 | 11 |
The Coordinate Plane
- The x-axis is horizontal; the y-axis is vertical
- They intersect at the origin (0, 0)
- An ordered pair (x, y) locates a point: go x units horizontally, then y units vertically
The Four Quadrants
| Quadrant | x-values | y-values | Example |
|---|---|---|---|
| I (top-right) | Positive | Positive | (3, 4) |
| II (top-left) | Negative | Positive | (−2, 5) |
| III (bottom-left) | Negative | Negative | (−3, −1) |
| IV (bottom-right) | Positive | Negative | (4, −2) |
Order Matters: (3, 5) and (5, 3) are different points. Always write (x, y) — the x-coordinate (horizontal) comes first.
Mathematics · Topic 1.2
Graphing Linear Functions
A linear function produces a straight-line graph. Understanding its equation helps you describe the line's steepness and position.
Slope-Intercept Form
y = mx + c where m = slope (gradient) and c = y-intercept
Understanding Slope (Gradient)
- Slope = riserun = change in ychange in x
- A positive slope means the line goes up from left to right
- A negative slope means the line goes down from left to right
- A slope of zero means a horizontal line
- The steeper the line, the larger the absolute value of the slope
The y-Intercept
The y-intercept is where the line crosses the y-axis. At this point, x = 0. In y = mx + c, the y-intercept is c.
Example: y = 2x + 3 crosses the y-axis at (0, 3).
How to Graph a Linear Function
- Plot the y-intercept (the point where x = 0)
- Use the slope to find a second point (rise over run from the y-intercept)
- Draw a straight line through both points, extending in both directions
Example: Graph y = 2x − 1
- y-intercept: (0, −1) — plot this point
- Slope = 2 = 2/1 — from (0, −1), go up 2 and right 1 to reach (1, 1)
- Draw a line through (0, −1) and (1, 1)
Common Mistake: Confusing slope and y-intercept. In y = 3x + 2, the slope is 3 (the coefficient of x) and the y-intercept is 2 (the constant term). Not the other way around!
Mathematics · Topic 1.2
Worked Examples
Study the full solutions below. Notice how each step is justified with a reason.
EXAMPLE 1Solve 4x − 7 = 13. Check your answer.
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Full Solution
Step 1: Add 7 to both sides: 4x = 20
Step 2: Divide both sides by 4: x = 5
Check: 4(5) − 7 = 20 − 7 = 13 ✓
Step 2: Divide both sides by 4: x = 5
Check: 4(5) − 7 = 20 − 7 = 13 ✓
EXAMPLE 2Solve −5x + 2 ≥ 22 and represent the solution on a number line.
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Full Solution
Step 1: Subtract 2 from both sides: −5x ≥ 20
Step 2: Divide both sides by −5. Since I’m dividing by a negative, I flip the inequality sign: x ≤ −4
Number line: Closed circle at −4 (because ≤ includes the boundary), shade to the left.
Solution set: All real numbers less than or equal to −4.
Step 2: Divide both sides by −5. Since I’m dividing by a negative, I flip the inequality sign: x ≤ −4
Number line: Closed circle at −4 (because ≤ includes the boundary), shade to the left.
Solution set: All real numbers less than or equal to −4.
EXAMPLE 3A function is defined as f(x) = 3x − 2. Find f(−3) and f(5).
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Full Solution
f(−3): Replace x with −3: f(−3) = 3(−3) − 2 = −9 − 2 = −11
f(5): Replace x with 5: f(5) = 3(5) − 2 = 15 − 2 = 13
f(5): Replace x with 5: f(5) = 3(5) − 2 = 15 − 2 = 13
EXAMPLE 4Solve 3x + 4 = x + 12.
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Full Solution
Step 1: Subtract x from both sides: 2x + 4 = 12
Step 2: Subtract 4 from both sides: 2x = 8
Step 3: Divide both sides by 2: x = 4
Check: Left side: 3(4) + 4 = 16. Right side: 4 + 12 = 16. ✓
Step 2: Subtract 4 from both sides: 2x = 8
Step 3: Divide both sides by 2: x = 4
Check: Left side: 3(4) + 4 = 16. Right side: 4 + 12 = 16. ✓
EXAMPLE 5State the slope and y-intercept of y = −2x + 5, and describe what the graph looks like.
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Full Solution
Comparing y = −2x + 5 with y = mx + c:
Slope (m): −2 — the line goes down from left to right (negative slope). For every 1 unit right, the line drops 2 units.
y-intercept (c): 5 — the line crosses the y-axis at (0, 5).
The graph is a straight line starting at (0, 5) and sloping downward to the right.
Slope (m): −2 — the line goes down from left to right (negative slope). For every 1 unit right, the line drops 2 units.
y-intercept (c): 5 — the line crosses the y-axis at (0, 5).
The graph is a straight line starting at (0, 5) and sloping downward to the right.
EXAMPLE 6A taxi charges $3 base fare plus $2 per kilometre. Write a function for the total cost and find the cost of a 7 km trip.
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Full Solution
Function: Let x = kilometres and C(x) = total cost.
C(x) = 2x + 3
For 7 km: C(7) = 2(7) + 3 = 14 + 3 = $17
The slope (2) represents the cost per kilometre. The y-intercept (3) is the base fare. This is a real-world application of a linear function.
C(x) = 2x + 3
For 7 km: C(7) = 2(7) + 3 = 14 + 3 = $17
The slope (2) represents the cost per kilometre. The y-intercept (3) is the base fare. This is a real-world application of a linear function.
EXAMPLE 7Is this a function? Input: {1, 2, 3, 2}, Output: {5, 8, 11, 9}. Explain.
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Full Solution
No, this is NOT a function.
The input 2 produces two different outputs (8 and 9). A function requires each input to have exactly one output. Since input 2 gives both 8 and 9, this relation fails the function test.
The input 2 produces two different outputs (8 and 9). A function requires each input to have exactly one output. Since input 2 gives both 8 and 9, this relation fails the function test.
Mathematics · Topic 1.2
Practice Q&A
Attempt each question before revealing the model answer. Show your working clearly.
SOLVESolve 7x − 3 = 25
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Model Answer
Add 3 to both sides: 7x = 28
Divide both sides by 7: x = 4
Check: 7(4) − 3 = 28 − 3 = 25 ✓
Divide both sides by 7: x = 4
Check: 7(4) − 3 = 28 − 3 = 25 ✓
SOLVESolve −2x + 8 > 14
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Model Answer
Subtract 8 from both sides: −2x > 6
Divide both sides by −2 (flip the sign!): x < −3
On a number line: open circle at −3, shade to the left.
Solution: all values less than −3.
Divide both sides by −2 (flip the sign!): x < −3
On a number line: open circle at −3, shade to the left.
Solution: all values less than −3.
EVALUATEIf f(x) = 4x − 5, find f(3) and f(−2).
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Model Answer
f(3) = 4(3) − 5 = 12 − 5 = 7
f(−2) = 4(−2) − 5 = −8 − 5 = −13
f(−2) = 4(−2) − 5 = −8 − 5 = −13
IDENTIFYIn which quadrant is the point (−4, 7)?
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Model Answer
x is negative and y is positive. This means the point is in Quadrant II (top-left). Quadrant II has negative x-values and positive y-values.
SOLVESolve 5x + 2 = 3x + 14
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Model Answer
Subtract 3x from both sides: 2x + 2 = 14
Subtract 2 from both sides: 2x = 12
Divide by 2: x = 6
Check: 5(6) + 2 = 32; 3(6) + 14 = 32 ✓
Subtract 2 from both sides: 2x = 12
Divide by 2: x = 6
Check: 5(6) + 2 = 32; 3(6) + 14 = 32 ✓
DESCRIBEDescribe the graph of y = −x + 4 (slope, y-intercept, direction).
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Model Answer
The equation is in slope-intercept form y = mx + c where m = −1 and c = 4.
Slope: −1 — the line goes down from left to right, dropping 1 unit for every 1 unit to the right.
y-intercept: (0, 4) — the line crosses the y-axis at 4.
The graph is a straight line starting at (0, 4) going downward to the right at a 45° angle.
Slope: −1 — the line goes down from left to right, dropping 1 unit for every 1 unit to the right.
y-intercept: (0, 4) — the line crosses the y-axis at 4.
The graph is a straight line starting at (0, 4) going downward to the right at a 45° angle.
APPLYA phone plan charges $15/month plus $0.10 per text message. Write a function for the monthly cost and find the cost for 200 texts.
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Model Answer
Let x = number of text messages, C(x) = monthly cost.
Function: C(x) = 0.10x + 15
For 200 texts: C(200) = 0.10(200) + 15 = 20 + 15 = $35
The slope (0.10) is the cost per text. The y-intercept (15) is the base monthly fee.
Function: C(x) = 0.10x + 15
For 200 texts: C(200) = 0.10(200) + 15 = 20 + 15 = $35
The slope (0.10) is the cost per text. The y-intercept (15) is the base monthly fee.
SOLVESolve 3(x + 4) = 21
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Model Answer
Expand: 3x + 12 = 21
Subtract 12: 3x = 9
Divide by 3: x = 3
Check: 3(3 + 4) = 3(7) = 21 ✓
Subtract 12: 3x = 9
Divide by 3: x = 3
Check: 3(3 + 4) = 3(7) = 21 ✓
Mathematics · Review
Flashcard Review
Tap each card to reveal the answer. Try to answer from memory first.
What is an inverse operation?
The operation that undoes another. Addition undoes subtraction; multiplication undoes division.
Tap to reveal
When do you flip the inequality sign?
When you multiply or divide both sides by a NEGATIVE number. Example: −2x < 6 becomes x > −3.
Tap to reveal
What is a function?
A rule that assigns exactly ONE output to each input. If one input gives two different outputs, it is NOT a function.
Tap to reveal
What does f(x) = 2x + 3 mean?
A function that takes input x, multiplies by 2, then adds 3 to produce the output.
Tap to reveal
What is the origin on a coordinate plane?
The point (0, 0) where the x-axis and y-axis intersect.
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In which quadrant are both x and y negative?
Quadrant III (bottom-left). Example: (−3, −5).
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What is the slope (gradient) of a line?
Rise ÷ run = (change in y) ÷ (change in x). It measures the steepness and direction of the line.
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In y = mx + c, what is c?
The y-intercept — the point where the line crosses the y-axis (where x = 0).
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Open circle vs closed circle on a number line?
Open circle (< or >) means the boundary is NOT included. Closed circle (≤ or ≥) means the boundary IS included.
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How do you solve a two-step equation?
Undo addition/subtraction first, then undo multiplication/division. Always do the same operation to both sides.
Tap to reveal
What is the solution set of an inequality?
All values of the variable that make the inequality true. It is usually a range of values, not a single number.
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Positive slope vs negative slope?
Positive slope: line goes UP from left to right. Negative slope: line goes DOWN from left to right.
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What is an ordered pair?
(x, y) — two values that locate a point on the coordinate plane. The first value is horizontal (x), the second is vertical (y).
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How do you check a solution to an equation?
Substitute the solution back into the original equation. Both sides should be equal.
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What does slope = 0 mean?
The line is perfectly horizontal (flat). The y-value never changes no matter what x is.
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Mathematics · Practice Test
Practice Test — 20 Questions
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