14.1 Algebraic Expressions, Patterns & Translating Verbal Phrases
Key Takeaways
Algebraic expressions are composed of variables, numerical coefficients, constant terms, and arithmetic operations; like terms must possess identical variable bases raised to identical exponents to be combined.
Evaluating expressions demands strict adherence to the Order of Operations (PEMDAS/GEMS), with particular care given to negative numbers: exponentiation of a negative base requires grouping parentheses, where (-x)^n != -x^n for even powers (e.g., (-3)² = 9 vs. -3² = -9).
Simplifying multi-step expressions requires applying the Distributive Property, correctly propagating negative signs across parentheses, expanding polynomial products (FOIL), and resolving nested grouping symbols from the innermost layer outward.
Numerical and geometric patterns are defined by explicit nth-term formulas: arithmetic sequences exhibit a constant common difference (a_n = a_1 + (n - 1)d), while geometric sequences exhibit a constant common ratio (a_n = a_1 · r^(n - 1)).
Translating verbal phrases into algebraic statements requires precision with turnaround phrases such as 'less than' and 'subtracted from' (e.g., '8 less than x' translates to x - 8, never 8 - x), alongside inequality constraints like 'is at least' (≥) and 'is no more than' (≤).
Algebraic Expressions, Patterns & Translating Verbal Phrases
Quick Answer: On the WEST-B Mathematics subtest (Objective 0017), algebraic competence requires fluency in three interconnected skills: manipulating formal expressions, extending numerical patterns, and translating word phrases into mathematical syntax. Remember: Like terms require identical variable bases and exponents (3x² and -5x² can combine, but 3x² and 3x cannot). When evaluating expressions with negative values, remember that (-4)² = 16 while -4² = -16. For patterns, use explicit formulas: Arithmetic Sequences (a_n = a_1 + (n - 1)d) with common difference d, and Geometric Sequences (a_n = a_1 · r^(n - 1)) with common ratio r. In translation, watch out for turnaround words: "7 less than y" is y - 7, and "is at least 15" translates to ≥ 15.
1. Algebraic Terminology & Structural Anatomy
Algebra uses symbolic notation to represent quantitative relationships, generalizations, and unknowns. To solve problems efficiently, you must understand its standard structural components.
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| ALGEBRAIC EXPRESSION ANATOMY |
| |
| 5x³ - 7x² + 2x - 19 |
| | | | | | | | |
| Coefficient --+ | | | | | +-- Constant Term (fixed value) |
| Variable --+ | | | +------- Linear Term (degree 1) |
| Quadratic -+ +------- Exponent / Power (degree 2) |
| Term |
| |
| +-------------------+----------------------------+------------------------------------------+ |
| | ELEMENT | FORMAL DEFINITION | EXAMPLE / IDENTIFICATION | |
| +-------------------+----------------------------+------------------------------------------+ |
| | Variable | Symbol (letter) for unknown| x, y, n, θ (represents varying values) | |
| +-------------------+----------------------------+------------------------------------------+ |
| | Constant | Fixed numerical value | -19, 4, 3/4, π (does not change) | |
| +-------------------+----------------------------+------------------------------------------+ |
| | Coefficient | Numerical multiplier | In -7x², the coefficient is -7 | |
| +-------------------+----------------------------+------------------------------------------+ |
| | Term | Single number, variable, | 5x³, -7x², 2x, and -19 are 4 distinct | |
| | | or product of both | terms separated by + or - | |
| +-------------------+----------------------------+------------------------------------------+ |
| | Like Terms | Terms with identical | 4ab² and -9ab² (can be combined); | |
| | | variable bases & exponents | 4a²b and 4ab² are UNLIKE (cannot combine)| |
| +-------------------+----------------------------+------------------------------------------+ |
| | Degree of Term | Sum of exponents on vars | 5x³ has degree 3; 6x²y³ has degree 2+3=5 | |
| +-------------------+----------------------------+------------------------------------------+ |
| | Degree of Poly | Highest degree of any term | 5x³ - 7x² + 2x - 19 has degree 3 (cubic) | |
| +-------------------+----------------------------+------------------------------------------+ |
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Types of Polynomials by Term Count
- Monomial: An algebraic expression containing exactly one term (e.g., -12x⁴, 7xy).
- Binomial: An algebraic expression containing exactly two terms connected by addition or subtraction (e.g., 3x² - 8, 5a + 2b).
- Trinomial: An algebraic expression containing exactly three terms (e.g., x² - 6x + 9).
- Polynomial: An algebraic expression composed of one or more terms with non-negative integer exponents.
2. Evaluating Algebraic Expressions & Order of Operations
Evaluating an expression means replacing every variable with its assigned numerical value and simplifying using the strict Order of Operations (PEMDAS / GEMS).
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| ORDER OF OPERATIONS HIERARCHY |
| |
| 1. G - Grouping Symbols Parentheses ( ), Brackets [ ], Braces { }, Fraction Bars, Radicals |
| 2. E - Exponents Powers, Roots, Absolute Values (|x|) |
| 3. M/D - Multiply/Divide Strictly LEFT-TO-RIGHT in order of appearance |
| 4. A/S - Add/Subtract Strictly LEFT-TO-RIGHT in order of appearance |
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The High-Yield Negative Base Pitfall
On the WEST-B exam, sign errors with exponents are among the most common mistakes:
- (-a)ⁿ (Parentheses Present): The negative sign is part of the base being multiplied. (-3)² = (-3) × (-3) = +9 (-2)³ = (-2) × (-2) × (-2) = -8 (-2)⁴ = (-2) × (-2) × (-2) × (-2) = +16
- -aⁿ (No Parentheses): The negative sign denotes multiplication by -1 after exponentiation. -3² = -(3 × 3) = -9 -2⁴ = -(2 × 2 × 2 × 2) = -16
Negative Base vs. Negative Coefficient Comparison
(-4)² = (-4) × (-4) = +16
-4² = -(4 × 4) = -16
(-4)³ = (-4) × (-4) × (-4) = -64
-4³ = -(4 × 4 × 4) = -64
Multi-Variable Rational Evaluation
When substituting negative values into rational expressions, evaluate the entire numerator and the entire denominator separately before dividing:
Example: Evaluate (3x² - 4xy + y²)/(2x - y) for x = -2, y = 3
- Numerator: 3(-2)² - 4(-2)(3) + (3)² = 3(4) - (-24) + 9 = 12 + 24 + 9 = 45
- Denominator: 2(-2) - 3 = -4 - 3 = -7
- Result: -45/7
3. Simplifying Algebraic Expressions & Polynomial Operations
Simplification produces an equivalent expression with the minimum number of terms and operations.
The Distributive Property
Multiplication distributes over addition and subtraction: a(b + c) = ab + ac a(b - c) = ab - ac
- Distributing a Negative Sign: Distributing a negative multiplier reverses every sign inside the parentheses: -(3x - 7y + 4) = -3x + 7y - 4 -5(2a - 4b + 1) = -10a + 20b - 5
Nested Grouping Symbols
When expressions contain parentheses ( ), brackets [ ], and braces { }, work from the innermost layer outward:
Simplify: 7 - 3[2x - 4(x - 3) + 5]
- Distribute the innermost -4: 2x - 4x + 12 + 5 = -2x + 17
- Substitute back into the brackets: 7 - 3[-2x + 17]
- Distribute the -3 across the brackets: 7 + 6x - 51
- Combine constants: 6x - 44
Multiplying Binomials (FOIL & Box Method)
To multiply two binomials (ax + b)(cx + d): (ax + b)(cx + d) = (ac·x²) [First] + (ad·x) [Outside] + (bc·x) [Inside] + (bd) [Last]
- Special Product 1 (Square of Binomial): (a + b)² = a² + 2ab + b² and (a - b)² = a² - 2ab + b²
- Special Product 2 (Difference of Squares): (a + b)(a - b) = a² - b²
4. Number & Geometric Patterns: Arithmetic & Geometric Sequences
A sequence is an ordered list of numbers following a systematic mathematical rule. On the WEST-B, questions test your ability to determine pattern rules, find missing terms, and calculate the n-th term.
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| SEQUENCE CLASSIFICATION & FORMULAS |
| |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | SEQUENCE TYPE | DEFINING CHARACTERISTIC | FORMULAS & PROPERTIES | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Arithmetic Sequence | Constant ADDITION/SUBTRACTION| Common difference: d = a_n - a_(n-1)| |
| | (Linear Growth) | between consecutive terms | Explicit: a_n = a_1 + (n - 1)d | |
| | | | Recursive: a_n = a_(n-1) + d | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Geometric Sequence | Constant MULTIPLICATION | Common ratio: r = a_n / a_(n-1) | |
| | (Exponential Growth) | by a fixed scalar ratio | Explicit: a_n = a_1 · r^(n - 1) | |
| | | | Recursive: a_n = a_(n-1) · r | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Quadratic Pattern | Second differences are | Explicit: a_n = An² + Bn + C | |
| | (Non-linear Growth) | constant (triangular numbers)| 2nd difference = 2A | |
| +-----------------------+-----------------------------+-------------------------------------+ |
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Arithmetic Sequence Step-by-Step Analysis
Consider the sequence: 4, 11, 18, 25, 32, ...
- Find the common difference d: 11 - 4 = 7, 18 - 11 = 7 => d = 7.
- Write the explicit formula: a_n = a_1 + (n - 1)d = 4 + (n - 1)7 = 4 + 7n - 7 = 7n - 3.
- Find the 40th term (a_40): a_40 = 7(40) - 3 = 280 - 3 = 277.
- Find which term equals 179: 7n - 3 = 179 => 7n = 182 => n = 26.
Geometric Sequence Step-by-Step Analysis
Consider the sequence: 6, -18, 54, -162, 486, ...
- Find the common ratio r: -18 / 6 = -3, 54 / (-18) = -3 => r = -3.
- Write the explicit formula: a_n = a_1 · r^(n - 1) = 6 · (-3)^(n - 1).
- Find the 7th term (a_7): a_7 = 6 · (-3)^(7 - 1) = 6 · (-3)⁶ = 6 · 729 = 4,374.
Geometric / Visual Tile Growth Patterns
Visual pattern questions present consecutive geometric stages (e.g., toothpicks, grid tiles, border stones):
- Stage 1: 4 squares
- Stage 2: 7 squares
- Stage 3: 10 squares
- Stage 4: 13 squares Since the increase per stage is constant (+3), it forms an arithmetic sequence with a_1 = 4 and d = 3. The rule for Stage n is a_n = 4 + (n - 1)3 = 3n + 1.
5. Translating Verbal Phrases into Expressions, Equations & Inequalities
Converting everyday English descriptions into algebraic symbolism is one of the most heavily tested skills on the WEST-B exam.
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| VERBAL TRANSLATION LEXICON |
| |
| +------------------+-----------------------------------------------+------------------------+ |
| | OPERATION | COMMON ENGLISH KEYWORDS / PHRASES | ALGEBRAIC TRANSLATION | |
| +------------------+-----------------------------------------------+------------------------+ |
| | Addition (+) | sum of, plus, increased by, more than, | Sum of x and 9 -> x + 9| |
| | | total of, exceeds by, combined | 6 more than k -> k + 6 | |
| +------------------+-----------------------------------------------+------------------------+ |
| | Subtraction (-) | difference, minus, decreased by, diminished by| Diff of x and 4 -> x - 4|
| | | **less than**, **subtracted from**, fewer than| 5 LESS THAN y -> y - 5 | |
| +------------------+-----------------------------------------------+------------------------+ |
| | Multiplication(·)| product of, times, twice (2x), triple (3x), | Product of 7 and w-> 7w| |
| | | of (fractions/percents: 30% of x -> 0.30x) | 3/4 of a number -> 3/4n| |
| +------------------+-----------------------------------------------+------------------------+ |
| | Division (÷) | quotient of, ratio of, divided by, per | Quotient of x and 8->x/8|
| +------------------+-----------------------------------------------+------------------------+ |
| | Equality (=) | is, equals, is equal to, results in, yields | 2x plus 3 IS 11 -> 2x+3=11|
| +------------------+-----------------------------------------------+------------------------+ |
| | At Least (≥) | is at least, minimum of, no less than, ≥ | x is at least 50 -> x≥50|
| +------------------+-----------------------------------------------+------------------------+ |
| | At Most (≤) | is at most, maximum of, no more than, ≤ | y is at most 20 -> y≤20| |
| +------------------+-----------------------------------------------+------------------------+ |
| | Greater (>) | strictly greater than, more than, exceeds | k exceeds 12 -> k > 12 | |
| +------------------+-----------------------------------------------+------------------------+ |
| | Less (<) | strictly less than, fewer than, below | p is under 10 -> p < 10| |
| +------------------+-----------------------------------------------+------------------------+ |
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The Critical "Turnaround Words" Trap
In standard English, some subtraction and comparison phrases invert the stated order of elements:
- "A decreased by B" => A - B (standard order)
- "A less B" => A - B (standard order)
- "A less than B" => B - A (Turnaround!)
- "A subtracted from B" => B - A (Turnaround!)
- "A fewer than B" => B - A (Turnaround!)
Example: "12 less than four times a number n" translates to: 4n - 12 (NEVER 12 - 4n).
Parenthetical Grouping Verbal Signals
- "The square of the sum of x and y" => (x + y)²
- "The sum of the squares of x and y" => x² + y²
- "Twice the difference of n and 5" => 2(n - 5)
- "The difference of twice n and 5" => 2n - 5
6. Step-by-Step Worked Problems & Exact Derivations
Problem 1: Complex Expression Evaluation with Nested Operations
Problem: Evaluate the expression (4a² - 3ab + 2b³)/(a - b) when a = -3 and b = -2.
Step-by-Step Solution:
- Substitute a = -3 and b = -2 with grouping parentheses into the numerator: Numerator = 4(-3)² - 3(-3)(-2) + 2(-2)³
- Calculate exponents first: (-3)² = 9 (-2)³ = -8 Numerator = 4(9) - 3(-3)(-2) + 2(-8)
- Perform multiplications left to right: 4(9) = 36 -3(-3)(-2) = 9(-2) = -18 => - (18) = -18 2(-8) = -16
- Combine numerator terms: Numerator = 36 - 18 - 16 = 18 - 16 = 2
- Evaluate denominator: Denominator = a - b = (-3) - (-2) = -3 + 2 = -1
- Divide numerator by denominator: Final Value = 2 / (-1) = -2
Problem 2: Sequence Analysis and Term Derivation
Problem: In an arithmetic sequence, the 4th term is 19 and the 11th term is 54. Determine the first term a_1, the common difference d, and calculate the value of the 35th term a_35.
Step-by-Step Solution:
- Set up a system using the arithmetic sequence formula a_n = a_1 + (n - 1)d:
- For n = 4: a_4 = a_1 + 3d = 19
- For n = 11: a_11 = a_1 + 10d = 54
- Subtract the first equation from the second to eliminate a_1: (a_1 + 10d) - (a_1 + 3d) = 54 - 19 7d = 35 => d = 5
- Substitute d = 5 into the first equation to find a_1: a_1 + 3(5) = 19 => a_1 + 15 = 19 => a_1 = 4
- Formulate the explicit rule: a_n = 4 + (n - 1)5 = 5n - 1
- Calculate the 35th term (a_35): a_35 = 5(35) - 1 = 175 - 1 = 174
Problem 3: Multi-Step Verbal Translation with Inequality Constraints
Problem: A community center is renting a venue that charges a flat reservation fee of $250 plus $18 per attendee. The catering service charges an additional $14 per attendee. If the total event budget cannot exceed $1,450, write an algebraic inequality to model this situation and determine the maximum number of attendees that can be hosted.
Step-by-Step Solution:
- Identify the variable: Let n represent the total number of attendees.
- Identify the fixed and variable costs:
- Flat reservation fee = 250
- Venue fee per attendee = 18n
- Catering cost per attendee = 14n
- Combined variable cost = 18n + 14n = 32n
- Translate 'cannot exceed': 'Cannot exceed' means ≤ (is less than or equal to).
- Formulate the inequality: 250 + 32n ≤ 1450
- Solve for n: 32n ≤ 1450 - 250 32n ≤ 1200 n ≤ 1200 / 32 = 37.5
- Interpret the result in context: Since attendees must be a whole integer, round down to the nearest whole number: Maximum Attendees = 37
What is the value of the algebraic expression 3x² - 2xy + y³ when evaluated at x = -2 and y = -3?
-27
-3
15
27
An arithmetic sequence begins with the terms 7, 13, 19, 25, ... What is the 25th term (a₂₅) of this sequence?
145
151
157
163
Which of the following algebraic statements correctly translates the verbal phrase: 'Five less than three times the square of the sum of a number k and 4 is at most 70'?
3k² + 4² - 5 < 70
5 - 3(k + 4)² ≤ 70
3(k + 4)² - 5 ≤ 70
3(k² + 4) - 5 ≥ 70
Which expression represents the complete simplification of 4[2x - 3(x - 5)] - 2(x + 8)?
2x + 44
-6x + 76
-2x + 44
-6x + 44
Sections you finish are checked off in the contents.