📓 Math Study Notebook
Linear Functions,
Equations & Inequalities
20 Core Problems · Examples · Memory Points · Self-Study Edition
Part 1
Linear Functions
SLOPE = RISE/RUN
y = mx + b
m = slope
b = y-intercept
PARALLEL → same m
PERPENDICULAR → m₁ × m₂ = −1
Slope formula: m = (y₂ − y₁) ÷ (x₂ − x₁). If m > 0 → going UP. If m < 0 → going DOWN. If m = 0 → FLAT line.
1
Example Problem
What is the slope of the line passing through the points \((2,\ 5)\) and \((6,\ 13)\)?
Step-by-Step
Use slope formula: \( m = \dfrac{y_2 - y_1}{x_2 - x_1} \)
\( m = \dfrac{13 - 5}{6 - 2} = \dfrac{8}{4} = \boxed{\ \ \ \ \ \ } \)
Always put y-values on TOP, x-values on BOTTOM!
\( m = \dfrac{13 - 5}{6 - 2} = \dfrac{8}{4} = \boxed{\ \ \ \ \ \ } \)
Always put y-values on TOP, x-values on BOTTOM!
Answer:
2
Practice
Find the slope of the line through \((-1,\ 4)\) and \((3,\ -4)\). Is this line going up or down? ⚡ TRICKY
Watch the signs! Subtracting negatives can flip your answer.
m = Direction:
3
Example Problem
Write the equation of a line with slope \(m = 3\) and y-intercept \(b = -2\).
Template
\( y = \underbrace{m}_{?} \cdot x + \underbrace{b}_{?} \)
Plug in \(m = 3\), \(b = -2\):
\( y = 3x + (-2) = \boxed{\ \ \ \ \ \ \ \ \ \ } \)
Plug in \(m = 3\), \(b = -2\):
\( y = 3x + (-2) = \boxed{\ \ \ \ \ \ \ \ \ \ } \)
Answer:
4
Practice — Commonly Missed!
A line passes through \((0,\ -3)\) and has slope \(\dfrac{2}{3}\). ⚠ COMMON MISTAKE
Which point is also on this line?
(a) \((3,\ -1)\) (b) \((3,\ 1)\) (c) \((6,\ 1)\) (d) \((2,\ -1)\)
Which point is also on this line?
(a) \((3,\ -1)\) (b) \((3,\ 1)\) (c) \((6,\ 1)\) (d) \((2,\ -1)\)
Start from (0, −3). Move right 3, up 2. Where do you land?
Answer:
5
Example Problem — Point-Slope Form
Find the equation of the line with slope \(-2\) passing through \((1,\ 4)\).
Point-Slope Formula
\( y - y_1 = m(x - x_1) \)
\( y - 4 = -2(x - 1) \)
\( y - 4 = -2x + 2 \)
\( y = \boxed{\ \ \ \ \ \ \ \ \ \ \ \ } \)
KEY: Use \( y - y_1 \) NOT \( y + y_1 \) !
\( y - 4 = -2(x - 1) \)
\( y - 4 = -2x + 2 \)
\( y = \boxed{\ \ \ \ \ \ \ \ \ \ \ \ } \)
KEY: Use \( y - y_1 \) NOT \( y + y_1 \) !
Final answer: y =
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Part 2
Linear Equations
ISOLATE the variable
DO SAME to BOTH sides
INVERSE operations
DISTRIBUTE first
COLLECT LIKE TERMS
CHECK your answer!
Order: ① Distribute → ② Combine like terms → ③ Move variables left, numbers right → ④ Divide
6
Example Problem — Basic Equation
Solve: \( 3x - 7 = 14 \)
Solution
\( 3x - 7 = 14 \)
\( 3x = 14 + 7 = 21 \) ← add 7 to both sides
\( x = \dfrac{21}{3} = \boxed{\ \ \ } \) ← divide both sides by 3
CHECK: \( 3(\boxed{\ \ \ }) - 7 = 14 \) ✓
\( 3x = 14 + 7 = 21 \) ← add 7 to both sides
\( x = \dfrac{21}{3} = \boxed{\ \ \ } \) ← divide both sides by 3
CHECK: \( 3(\boxed{\ \ \ }) - 7 = 14 \) ✓
x =
7
Practice — Distributive Property
Solve: \( 2(x + 3) = 4x - 6 \) ⚡ TRICKY
Distribute the 2 first: 2·x + 2·3. Then move x terms to one side!
x =
8
Example Problem — Variables on Both Sides
Solve: \( 5x + 4 = 2x + 13 \)
Step-by-Step
\( 5x + 4 = 2x + 13 \)
\( 5x - 2x = 13 - 4 \) ← x LEFT, numbers RIGHT
\( 3x = 9 \)
\( x = \boxed{\ \ \ } \)
\( 5x - 2x = 13 - 4 \) ← x LEFT, numbers RIGHT
\( 3x = 9 \)
\( x = \boxed{\ \ \ } \)
x =
9
Practice — Fraction Equations ⚠ MOST MISSED
Solve: \( \dfrac{x}{3} + 2 = 5 \)
Multiply BOTH sides by 3 first to clear the fraction. Then solve normally.
x =
10
Example Problem — Special Cases
Solve: \( 3(x - 2) = 3x - 6 \) ⚡ INFINITE SOLUTIONS?
Watch Out!
\( 3x - 6 = 3x - 6 \) ← after distributing
\( 3x - 3x = -6 + 6 \)
\( 0 = 0 \) ← TRUE! → Infinite Solutions (All real numbers)
If you get \( 0 = 5 \) (FALSE) → No Solution!
\( 3x - 3x = -6 + 6 \)
\( 0 = 0 \) ← TRUE! → Infinite Solutions (All real numbers)
If you get \( 0 = 5 \) (FALSE) → No Solution!
Answer:
🔑 "No solution" = contradiction (0 = 5)
"Infinite solutions" = identity (0 = 0)
"One solution" = variable remains (x = #)
"Infinite solutions" = identity (0 = 0)
"One solution" = variable remains (x = #)
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Part 3
Linear Inequalities
FLIP SIGN when ÷ or × by NEGATIVE
OPEN circle = < or >
CLOSED circle = ≤ or ≥
SHADE LEFT = smaller values
SHADE RIGHT = larger values
⚠ THE BIG RULE: Multiply or divide both sides by a NEGATIVE number → FLIP the inequality sign!
11
Example Problem — Basic Inequality
Solve and graph: \( x + 3 > 7 \)
Solution + Number Line
\( x + 3 > 7 \)
\( x > 7 - 3 \)
\( x > \boxed{\ \ \ } \)
Graph: ◦————————→ (open circle, shade right)
> means OPEN circle. ≥ means CLOSED circle ●
\( x > 7 - 3 \)
\( x > \boxed{\ \ \ } \)
Graph: ◦————————→ (open circle, shade right)
> means OPEN circle. ≥ means CLOSED circle ●
x >
12
Example Problem — FLIP THE SIGN! ⚠ #1 MISTAKE
Solve: \( -2x < 8 \) ⚡ SIGN FLIP!
Critical Step
\( -2x < 8 \)
Divide both sides by \(-2\): FLIP the sign!
\( x \; \boxed{?} \; -4 \)
❌ Wrong: \( x < -4 \)
✅ Right: \( x > -4 \) ← sign flipped from < to >
Divide both sides by \(-2\): FLIP the sign!
\( x \; \boxed{?} \; -4 \)
❌ Wrong: \( x < -4 \)
✅ Right: \( x > -4 \) ← sign flipped from < to >
Answer: x
13
Practice — Two-Step Inequality
Solve: \( 3x - 5 \leq 10 \). Then graph on the number line below.
Number Line Space
←——|——|——|——|——|——|——|——→
-2 -1 0 1 2 3 4 5
-2 -1 0 1 2 3 4 5
Answer: x ≤
14
Example Problem — Compound Inequality
Solve: \( -1 \leq 2x + 3 \leq 9 \)
Split into Parts
Subtract 3 from ALL parts:
\( -1 - 3 \leq 2x \leq 9 - 3 \)
\( -4 \leq 2x \leq 6 \)
Divide ALL parts by 2:
\( \boxed{\ \ } \leq x \leq \boxed{\ \ } \)
Do the SAME operation to ALL THREE parts!
\( -1 - 3 \leq 2x \leq 9 - 3 \)
\( -4 \leq 2x \leq 6 \)
Divide ALL parts by 2:
\( \boxed{\ \ } \leq x \leq \boxed{\ \ } \)
Do the SAME operation to ALL THREE parts!
Answer: ≤ x ≤
15
Practice — Tricky Compound ⚡ TRICKY
Solve and write in interval notation: \( 2 < -x + 5 \leq 8 \) ⚠ NEGATIVE x!
Multiply ALL parts by −1 at the end → FLIP BOTH inequality signs!
Answer:
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Part 4
Mixed & Word Problems
DEFINE variable first
TRANSLATE words → math
AT LEAST = ≥
AT MOST = ≤
MORE THAN = >
LESS THAN = <
"is" → = | "more than" → + | "less than" → − | "times" → × | "per" → ÷
16
Example Problem — Word to Equation
Five more than twice a number is 19. Find the number.
Translation Step
Let \( n \) = the unknown number
"twice a number" → \( 2n \)
"five more than" → \( 2n + 5 \)
"is 19" → \( = 19 \)
Equation: \( 2n + 5 = 19 \)
\( 2n = 14 \)
\( n = \boxed{\ \ \ } \)
"twice a number" → \( 2n \)
"five more than" → \( 2n + 5 \)
"is 19" → \( = 19 \)
Equation: \( 2n + 5 = 19 \)
\( 2n = 14 \)
\( n = \boxed{\ \ \ } \)
n =
17
Practice — Word to Inequality
A student needs at least 90 points to get an A. She already has 78 points.
How many more points \((p)\) does she need?
Write and solve the inequality.
How many more points \((p)\) does she need?
Write and solve the inequality.
"At least 90" means the total must be ≥ 90. Total = 78 + p.
Inequality: Answer: p ≥
18
Example Problem — Slope as Rate of Change
A taxi costs \$3.00 base fare plus \$2.50 per mile. Write a linear equation for the total cost \(C\) after \(m\) miles. How much does a 6-mile ride cost?
Build the Equation
Base fare (y-intercept) \(b = 3.00\)
Cost per mile (slope) \(m = 2.50\)
Equation: \( C = \boxed{\ \ \ \ \ } m + \boxed{\ \ \ \ \ } \)
For 6 miles: \( C = 2.50(6) + 3.00 = \$ \boxed{\ \ \ \ } \)
Cost per mile (slope) \(m = 2.50\)
Equation: \( C = \boxed{\ \ \ \ \ } m + \boxed{\ \ \ \ \ } \)
For 6 miles: \( C = 2.50(6) + 3.00 = \$ \boxed{\ \ \ \ } \)
6-mile cost: $
19
Practice — Graphs & Equations ⚡ VISUAL TRAP
A line crosses the y-axis at \(4\) and the x-axis at \(-2\).
(a) Find the slope. (b) Write the equation. (c) Is \((4,\ 12)\) on this line?
(a) Find the slope. (b) Write the equation. (c) Is \((4,\ 12)\) on this line?
x-intercept means y = 0. So you have two points: (0, 4) and (−2, 0). Use slope formula!
(c) On the line? Yes / No — because:
20
Challenge — Everything Combined ⚠ BOSS LEVEL
Two friends start walking toward each other.
Alex starts at position \(x = 0\) and walks at \(4\) ft/sec.
Jordan starts at position \(x = 60\) and walks at \(-3\) ft/sec.
(a) Write equations for each person's position over time \(t\).
(b) Find when they meet (set equations equal).
(c) Where do they meet on the number line?
Alex starts at position \(x = 0\) and walks at \(4\) ft/sec.
Jordan starts at position \(x = 60\) and walks at \(-3\) ft/sec.
(a) Write equations for each person's position over time \(t\).
(b) Find when they meet (set equations equal).
(c) Where do they meet on the number line?
Setup Guide
Alex: \( A(t) = 0 + 4t = 4t \)
Jordan: \( J(t) = 60 + (-3t) = \boxed{\ \ \ \ \ \ \ \ } \)
They meet when \( A(t) = J(t) \):
\( 4t = \boxed{\ \ \ \ \ \ \ \ \ \ } \)
\( t = \boxed{\ \ \ } \) seconds
Position: \( A(\boxed{\ \ \ }) = \boxed{\ \ \ \ \ } \) ft from start
Jordan: \( J(t) = 60 + (-3t) = \boxed{\ \ \ \ \ \ \ \ } \)
They meet when \( A(t) = J(t) \):
\( 4t = \boxed{\ \ \ \ \ \ \ \ \ \ } \)
\( t = \boxed{\ \ \ } \) seconds
Position: \( A(\boxed{\ \ \ }) = \boxed{\ \ \ \ \ } \) ft from start
Quick Ref
🔑 Key Formulas Cheat Sheet
SLOPE
\( m = \dfrac{y_2 - y_1}{x_2 - x_1} \)
\( m = \dfrac{\text{rise}}{\text{run}} \)
\( m = \dfrac{\text{rise}}{\text{run}} \)
EQUATIONS
Slope-intercept: \( y = mx + b \)
Point-slope: \( y - y_1 = m(x - x_1) \)
Point-slope: \( y - y_1 = m(x - x_1) \)
INEQUALITY RULES
Multiply/divide by NEGATIVE → FLIP sign
\( < \) or \( > \) → OPEN ○
\( \leq \) or \( \geq \) → CLOSED ●
\( < \) or \( > \) → OPEN ○
\( \leq \) or \( \geq \) → CLOSED ●
WORD PROBLEM KEYS
at least → \( \geq \)
at most → \( \leq \)
more than → \( > \)
less than → \( < \)
at most → \( \leq \)
more than → \( > \)
less than → \( < \)
MY NOTES & REFLECTIONS
Things I found easy: _______________________________________
Things I'm still confused about: ____________________________
My strategy to improve: ____________________________________
Things I'm still confused about: ____________________________
My strategy to improve: ____________________________________