7.1 Basic Algebra, Linear Equations & Inequalities

Basic Algebra, Linear Equations & Inequalities

Algebraic reasoning serves as the universal symbolic language of competitive quantitative examinations. On the NAPOLCOM PNP Entrance Examination (PNPE), algebra is not tested as an abstract theoretical branch of mathematics, but as a practical analytical tool for modeling administrative scenarios, logistical inventories, personnel shifts, and resource constraints under strict calculator-free conditions. Developing speed and fluency in algebraic manipulation ensures that candidates can transform wordy narrative prompts into concise equations and solve them rapidly without computational error.

Independent Preparation Notice: This study module is independently developed by OpenExamPrep to assist prospective applicants in mastering the quantitative reasoning concepts required for the examination. OpenExamPrep is an independent educational publisher and is not affiliated with, endorsed by, or partnered with the National Police Commission (NAPOLCOM) or the Philippine National Police (PNP).

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1. Foundational Algebraic Reasoning on the Calculator-Free PNPE

Unlike secondary school examinations where calculators are routinely permitted, the NAPOLCOM PNPE tests foundational mental calculation and structural algebraic comprehension. The strict prohibition of computing aids means that every algebraic step—clearing parentheses, factoring quadratics, balancing linear equations, and dividing polynomials—must be executed manually with speed and precision.

Core Terminology and Algebraic Hierarchy

• Constant: A fixed numerical quantity whose value never changes (e.g., 7, -12, 3/4, π).
• Variable: A symbol (typically an English or Greek letter such as x, y, z, t, θ) representing an unknown or changing quantity.
• Term: A single numerical constant, a single variable, or the algebraic product of constants and variables separated from other terms by addition (+) or subtraction (-). For example, in the expression 5x² - 3xy + 8, the terms are 5x², -3xy, and 8.
• Numerical Coefficient: The multiplicative numerical factor preceding a variable. In the term -7x²y, the numerical coefficient is -7.
• Literal Coefficient: The variable component of a term, including its exponents. In -7x²y, the literal coefficient is x²y.
• Like Terms (Similar Terms): Terms that possess the exact same literal coefficients raised to the exact same powers, regardless of their numerical coefficients. For instance, 4x²y and -9x²y are like terms and can be combined by adding their numerical coefficients: 4x²y + (-9x²y) = -5x²y. Conversely, 3x²y and 3xy² are unlike terms and cannot be combined into a single monomial.

Fundamental Axioms of Real Numbers in Algebra

Three foundational algebraic properties govern all expression manipulation and simplification:

1. Commutative Property:
   • Addition: a + b = b + a
   • Multiplication: a × b = b × a (Note: Subtraction and division are not commutative; a - b ≠ b - a and a ÷ b ≠ b ÷ a when a ≠ b.)
2. Associative Property:
   • Addition: (a + b) + c = a + (b + c)
   • Multiplication: (a × b) × c = a × (b × c)
3. Distributive Property of Multiplication over Addition / Subtraction:
   • a(b + c) = ab + ac
   • a(b - c) = ab - ac

The FOIL Method for Binomial Multiplication

When multiplying two binomials (a + b)(c + d), apply the FOIL acronym (First, Outer, Inner, Last):

• First terms: a × c
• Outer terms: a × d
• Inner terms: b × c
• Last terms: b × d

For example, multiplying (2x - 3)(4x + 5):

• First: (2x)(4x) = 8x²
• Outer: (2x)(5) = +10x
• Inner: (-3)(4x) = -12x
• Last: (-3)(5) = -15
• Combine inner and outer terms: 8x² + 10x - 12x - 15 = 8x² - 2x - 15

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2. Polynomial Mechanics & Factoring Paradigms

Factoring is the mathematical process of decomposing an algebraic sum or difference into a product of simpler multiplicative expressions. Factoring enables rapid algebraic fraction simplification and root determination without resorting to the quadratic formula.

Paradigm 1: Common Monomial Factoring (CMF)

Common Monomial Factoring represents the distributive property applied in reverse. To factor by CMF:

1. Determine the Greatest Common Factor (GCF) of all numerical coefficients.
2. Identify the common literal variables present in every term, taking each variable to its lowest exponent appearing in the polynomial.
3. Divide each term of the polynomial by the identified monomial factor.

Example: Factor 18x³y² - 24x²y³ + 12x²y.

• The GCF of numerical coefficients 18, 24, and 12 is 6.
• The common variables are x (lowest power is x²) and y (lowest power is y¹).
• The common monomial factor is 6x²y.
• Factored expression: 6x²y(3xy - 4y² + 2).

Paradigm 2: Difference of Two Squares

The difference of two perfect squares factors into the product of conjugate binomials:

a² - b² = (a - b)(a + b)

Critical Exam Rule: The sum of two squares, a² + b², is prime over the set of real numbers and cannot be factored into real linear factors.

Rapid Mental Math Application of Difference of Squares

Under timed testing conditions, recognizing the difference of two squares converts cumbersome arithmetic multiplications into instant mental subtractions and additions:

Calculate: 78² - 22²

• Standard Method: Multiply 78 × 78 = 6,084 and 22 × 22 = 484, then subtract: 6,084 - 484 = 5,600 (lengthy and prone to borrowing errors).
• Algebraic Factoring Shortcut: Apply a² - b² = (a - b)(a + b): 78² - 22² = (78 - 22)(78 + 22) = 56 × 100 = 5,600 The computation is resolved mentally in three seconds.

Paradigm 3: Factoring Quadratic Trinomials (ax² + bx + c)

Case A: Simple Trinomials where a = 1 (x² + bx + c)

To factor x² + bx + c, find two integers p and q that satisfy two simultaneous conditions:

1. p × q = c (their product equals the constant term)
2. p + q = b (their sum equals the middle linear coefficient)

Once p and q are identified, the trinomial factors directly into (x + p)(x + q).

Example: Factor x² - 9x + 20.

• Identify factor pairs of +20 whose sum is -9: (-4) × (-5) = +20, and (-4) + (-5) = -9.
• Factored form: (x - 4)(x - 5).

Case B: General Trinomials where a ≠ 1 (The AC Grouping Method)

When the leading coefficient a is not 1, factoring by inspection can lead to trial-and-error delays. The AC Grouping Method guarantees a systematic solution:

1. Calculate the product of the leading coefficient and the constant term: a × c.
2. Find two integers p and q such that p × q = ac and p + q = b.
3. Rewrite the middle term bx as the sum px + qx.
4. Factor the resulting four-term polynomial by grouping in pairs.

Example: Factor 3x² + 10x - 8.

• Step 1: ac = 3 × (-8) = -24.
• Step 2: Find factors of -24 that sum to +10: 12 × (-2) = -24, and 12 + (-2) = +10.
• Step 3: Split the middle term: 3x² + 12x - 2x - 8.
• Step 4: Group in pairs: [3x² + 12x] - [2x + 8] = 3x(x + 4) - 2(x + 4) = (3x - 2)(x + 4).

Simplifying Rational Algebraic Expressions

To simplify algebraic fractions, factor the numerator and denominator completely and cancel identical non-zero factors:

(2x² - 8) / (x² + 5x + 6) = [2(x² - 4)] / [(x + 2)(x + 3)] = [2(x - 2)(x + 2)] / [(x + 2)(x + 3)] = 2(x - 2) / (x + 3) (where x ≠ -2, -3).

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3. Solving Single-Variable Linear Equations Step-by-Step

A linear equation in one variable is an equation that can be expressed in the form ax + b = 0, where a and b are real numbers and a ≠ 0. The exponent of the variable is strictly 1.

The Golden Principle of Equation Balancing

An algebraic equation functions like a precision balance scale. Whatever operation is executed on one side must be identically executed on the opposite side to maintain truth and equality:

• Addition Property: If A = B, then A + c = B + c.
• Subtraction Property: If A = B, then A - c = B - c.
• Multiplication Property: If A = B, then A × c = B × c.
• Division Property: If A = B, then A ÷ c = B ÷ c (provided c ≠ 0).

The Standard 4-Step Solution Protocol

To solve any single-variable linear equation systematically without misplacing signs or terms, execute the following sequence:

1. Step 1: Clear Fractions and Decimals: Multiply every single term on both sides of the equation by the Least Common Denominator (LCD) of all fractions present.
2. Step 2: Clear Grouping Symbols: Apply the distributive property to eliminate all parentheses, brackets, and braces. Be especially vigilant when distributing negative signs.
3. Step 3: Transpose and Combine Like Terms: Collect all terms containing the variable on one side of the equation (conventionally the left side) and all constant numbers on the opposite side, using inverse addition or subtraction.
4. Step 4: Isolate the Variable: Divide both sides of the equation by the numerical coefficient of the variable to obtain the final single-unit solution (x = k).

Detailed Worked Problem with Fractional Terms and Negative Distribution

Solve for x in the equation:

(3x - 2)/5 - (x - 4)/3 = (x + 6)/15 + 1

• Step 1 (Clear Denominators): The denominators are 5, 3, and 15. Their LCD is 15. Multiply every term on both sides by 15: 15 × [(3x - 2)/5] - 15 × [(x - 4)/3] = 15 × [(x + 6)/15] + 15 × (1) Simplify each term: 3(3x - 2) - 5(x - 4) = 1(x + 6) + 15
• Step 2 (Clear Grouping Symbols): Distribute 3 and -5 carefully: 9x - 6 - 5x + 20 = x + 6 + 15 (Notice that -5 × -4 = +20. Forgetting to reverse this sign is the most frequent candidate trap!)
• Step 3 (Combine Like Terms on Each Side): (9x - 5x) + (-6 + 20) = x + 21 4x + 14 = x + 21
• Step 4 (Transpose Terms): Subtract x from both sides: 3x + 14 = 21. Subtract 14 from both sides: 3x = 7.
• Step 5 (Isolate Variable): x = 7/3 = 2 1/3

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4. Simultaneous Linear Equations: Systems of Two Equations in Two Variables

A system of linear equations consists of two or more linear equations containing the same set of variables. Systems involving two equations in two unknowns can model logistical inventory distributions, ticket sales, vehicle types, and equipment allocations in quantitative problem-solving practice.

Mathematical Structure of a 2-Variable System

• Equation 1: a₁x + b₁y = c₁
• Equation 2: a₂x + b₂y = c₂

Geometric Interpretation of Solutions

1. Independent and Consistent (Unique Solution): The two lines have different slopes and intersect at exactly one coordinate point (x, y). This point satisfies both equations simultaneously.
2. Inconsistent (No Solution): The two lines have identical slopes but different y-intercepts. The lines are strictly parallel and never intersect (solution set is empty: ∅).
3. Dependent and Consistent (Infinitely Many Solutions): The two equations represent the exact same line with identical slopes and y-intercepts. Every point on the line is a valid solution.

Method 1: The Substitution Method

The substitution method is most effective when at least one variable in either equation has a coefficient of +1 or -1.

1. Isolate one variable with coefficient ±1 in terms of the other variable.
2. Substitute the resulting expression into the other equation, creating a single-variable linear equation.
3. Solve for the single variable.
4. Back-substitute the numerical value into the isolated expression to determine the second variable.

Method 2: The Elimination (Linear Combination) Method

The elimination method is superior when all variable coefficients are integers greater than 1.

1. Multiply one or both equations by non-zero constants so that the coefficients of one chosen variable become exact additive inverses (opposite signs, equal magnitude, e.g., +6y and -6y).
2. Add the two equations vertically. The chosen variable is eliminated (+6y + (-6y) = 0).
3. Solve the resulting single-variable equation.
4. Substitute the solved value back into either original equation to find the other variable.

Comprehensive Law Enforcement Applied Inventory Problem

A police station logistics officer conducts an audit of an impound facility containing 75 motor vehicles consisting entirely of patrol motorcycles (2 wheels each) and patrol utility vans (4 wheels each). An inspection confirms that the total number of mounted wheels across all 75 vehicles is 210 wheels (excluding spare tires). How many patrol motorcycles and how many patrol utility vans are stored in the facility?

Algebraic Modeling

• Let m = number of patrol motorcycles.
• Let v = number of patrol utility vans.

Set up the system of two simultaneous equations:

1. Equation 1 (Total Vehicle Count): m + v = 75
2. Equation 2 (Total Wheel Count): 2m + 4v = 210

Solution via Elimination Method

• Multiply Equation 1 by -2 to eliminate m: -2(m + v) = -2(75) ⟹ -2m - 2v = -150
• Add this scaled equation directly to Equation 2: (-2m - 2v) + (2m + 4v) = -150 + 210 0m + 2v = 60
• Solve for v: 2v = 60 ⟹ v = 30 utility vans
• Back-substitute v = 30 into Equation 1: m + 30 = 75 ⟹ m = 75 - 30 = 45 motorcycles

Verification

• Total vehicles: 45 + 30 = 75 (matches condition 1).
• Total wheels: (45 × 2) + (30 × 4) = 90 + 120 = 210 wheels (matches condition 2).

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5. Linear Inequalities in One Variable & The Negative Inversion Rule

A linear inequality describes a relationship between two expressions that are not strictly equal, using the inequality relations:

• < ("strictly less than")
• ≤ ("less than or equal to", "at most", "does not exceed")

• ("strictly greater than")

• ≥ ("greater than or equal to", "at least", "no less than")

The Cardinal Rule of Inequalities: Negative Multiplicative Inversion

While linear inequalities follow the same balancing steps as linear equations (adding or subtracting terms across sides does not affect the inequality relation), there is one crucial mathematical exception:

The Cardinal Rule: Whenever both sides of an inequality are multiplied or divided by a negative number, the direction of the inequality symbol MUST BE REVERSED (< becomes >, ≤ becomes ≥, > becomes <, and ≥ becomes ≤).

Mathematical Demonstration

Consider the true numerical inequality: 3 < 7.

• If we multiply both sides by positive 2: 3(2) < 7(2) ⟹ 6 < 14 (Truth is preserved).
• If we multiply both sides by negative 2 without reversing: 3(-2) < 7(-2) ⟹ -6 < -14.
  • On the real number line, -6 is located to the right of -14, meaning -6 is strictly greater than -14.
  • The statement -6 < -14 is completely false.
  • Reversing the symbol yields -6 > -14, preserving mathematical truth.

Number Line Representation and Interval Notation

Worked Step-by-Step Inequality Drill

Determine the complete solution set for the linear inequality:

7 - 3(2x - 5) ≥ 40

• Step 1 (Distribute): 7 - 6x + 15 ≥ 40
• Step 2 (Combine constants on the left): 22 - 6x ≥ 40
• Step 3 (Transpose constant term): Subtract 22 from both sides: -6x ≥ 40 - 22 ⟹ -6x ≥ 18
• Step 4 (Divide by negative coefficient and reverse inequality symbol): x ≤ 18 / (-6) ⟹ x ≤ -3
• Interval Notation: (-∞, -3].

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6. Translating Complex English Narrative Constraints into Algebraic Equations

The primary barrier to scoring high on civil service math is not calculating equations, but translating narrative English text into formal mathematical syntax. Candidates must recognize linguistic keywords and map them directly to mathematical operators.

The Verbal-to-Symbolic Translation Dictionary

Phrasing Traps in Verbal Translation

1. The "Less Than" Order Reversal Trap:
   • Prompt: "Seven less than three times a number x."
   • Incorrect Translation: 7 - 3x.
   • Correct Translation: 3x - 7.
2. The Grouping Trap (Square of Sum vs. Sum of Squares):
   • Square of the sum of x and y: First add, then square ⟹ (x + y)².
   • Sum of the squares of x and y: Square individually, then add ⟹ x² + y².
   • Note that (x + y)² = x² + 2xy + y² ≠ x² + y².
3. Consecutive Integer Modeling:
   • Consecutive Integers: x, x + 1, x + 2, x + 3, ...
   • Consecutive Even Integers: x, x + 2, x + 4, x + 6, ... (where x is even)
   • Consecutive Odd Integers: x, x + 2, x + 4, x + 6, ... (where x is odd) (Trap: Candidates frequently write x, x + 1, x + 3 for consecutive odd integers. Odd numbers are separated by intervals of 2, so the algebraic step remains +2!)

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7. High-Pressure Algebraic Error Avoidance Checklist

Before confirming an answer on the optical mark sheet, cross-check your scratchpad derivation against this 5-point checklist:

• Negative Sign Distribution: Did you distribute the negative sign to all terms inside the parentheses? (e.g., -(2x - 7) = -2x + 7, not -2x - 7).
• Fractions Cleared Completely: When multiplying by the LCD, did you remember to multiply stand-alone whole numbers and constants on both sides?
• Inequality Inversion Check: Did you divide or multiply by a negative number in the final step? If so, did you flip the inequality sign?
• Verification of What the Question Asks: Did the question ask for x, or did it ask for 2x + 1, the larger number, or the total inventory?
• Substituted Solution Test: Does your solved value satisfy the original prompt's constraints when plugged back into the initial narrative?