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Inequalities:
Number line conventions:
Use open circles for < and >
Use closed circles for ≤ and ≥
Example: –3 ≤ x < 1 → Closed circle at –3, open circle at 1
Solving Linear Inequalities
Steps:
Treat the inequality like an equation
Solve for the variable
Flip the inequality sign if you multiply or divide by a negative
Example 1: 3x < 2x + 4
→ Subtract 2x so x < 4
Example 2: –3 ≤ 3x – 2 < 7
Solve each part separately:
–3 ≤ 3x – 2 → 3x ≥ –1 → x ≥ –1⁄3
3x – 2 < 7 → 3x < 9 → x < 3
Final Answer: –1⁄3 ≤ x < 3
Graphical Representation of Inequalities
When graphing inequalities in two variables:
Use solid lines for ≤ or ≥
Use broken lines for < or >
Shade the unwanted region unless told otherwise
Example:
Graph the region defined by:
x < 1 → broken vertical line
y ≥ 1 → solid horizontal line
Shade the region that does not satisfy the inequalities
Sequences:
Recognizing and Continuing Sequences
What to do:
Identify the pattern or rule
Continue the sequence by applying the rule
Example:
Sequence: 2, 4, 6, 8, …
Rule: Add 2 Next terms: 10, 12
Sequence: 1, 3, 6, 10, 15, …
Rule: Add the difference between the integers (2, 3, 4, 5, …) Next terms: 21, 28
Term-to-Term Rule
Definition: The rule that tells you how to get from one term to the next.
Example: Sequence: 5, 10, 20, 40, … Rule: Multiply by 2
Finding the nth Term
You may be asked to find a formula for the nth term of a sequence.
Linear Sequences
Form: Tₙ = an + b Where a is the common difference, n is the number of the term, and b is the starting value.
Example:
Sequence: 3, 5, 7, 9, …
Common difference: +2 so the formula is: Tₙ = 2n + 1
If you wanted the 6th term: 2(6) + 1 = 12 + 1 = 13
Quadratic Sequences:
General:
Form: Tₙ = an² + bn + c
Used when the second difference is constant (the difference between the numbers in the first difference).
Example:
Sequence: 2, 6, 12, 20, …
First differences: +4, +6, +8
Second differences : +2 This suggests a quadratic rule.
Finding the formula:
Since it comes in the form Tₙ = an² + bn + c, you have to remember these:
2a = second difference
3a + b = first difference
a + b + c = first term
Now, to find the equation of the previous sequence: 2, 6, 12, 20, ...
2a = 2 therefore a = 1
3(1) + b = 4 so b = 1
1 + 1 + c = 2 so c = 0
Formula: Tₙ = n² + n
Cubic Sequences:
General
Form: Tₙ = an³ + bn² + cn + d
Used when the third difference is constant.
Example:
Sequence: 1, 8, 27, 64, 125, …
First difference: 7, 19, 37, 61
Second difference: 12, 18, 24
Third difference: 6, 6
Finding the formula:
Since it comes in the form Tₙ = an³ + bn² + cn + d, you have to remember these:
6a = third difference
12a + 2b = second difference
7a + 3b + c = first difference
a + b + c + d = first term
Now, to find the equation of the previous sequence: 1, 8, 27, 64, 125, ...
6a = 6 so a = 1
12 + 2b = 12 so b = 0
7 + c = 7 so c = 0
1 + d = 1 so d = 0
Rule: n³
Subscript Notation:
Tₙ represents the nth term of a sequence
Used in formulas and explanations
Example:
If Tₙ = 2n + 3, then:
T₁ = 5 since 2(1) + 3 = 5
T₂ = 7 since 2(2) + 3 = 7
T₃ = 9 since 2(3) + 3 = 9
NOTES DONE BY FARIDA SABET
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