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6.2 High School Algebra: Linear Equations, Inequalities & Polynomials
Armed Forces Classification Test AFCT
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1 Chapter 1: Guide Scope, AFQT & Service-Specific Policy
1.1 Editorial Scope and Service-Specific AFCT Policy 1.2 AFQT Meaning and Study Priorities 1.3 Service Composites and Targeted Skill Planning 1.4 Format-Agnostic Test Preparation and Pacing
2 Chapter 2: General Science (GS)
2.1 Life Science: Cell Biology, Human Anatomy & Genetics 2.2 Physical Science: Matter, Energy, Forces & Motion 2.3 Earth & Space Science: Geology, Meteorology & Astronomy 2.4 Chemistry Fundamentals: Atomic Structure, Elements & Chemical Reactions
3 Chapter 3: Arithmetic Reasoning (AR)
3.1 Word Problems: Rates, Ratios, Proportions & Percentages 3.2 Financial & Distance Math: Interest, Profit/Loss, Speed & Time 3.3 Work & Mixture Problems: Joint Rates & Percentage Concentrations 3.4 Multi-Step Problem Solving & Algebraic Word Problems
4 Chapter 4: Word Knowledge (WK)
4.1 General Academic & Technical Vocabulary 4.2 Context Clues, Synonyms & Antonyms 4.3 Word Roots, Prefixes & Suffixes (Etymology) 4.4 Advanced Technical & Operational Terminology
5 Chapter 5: Paragraph Comprehension (PC)
5.1 Identifying Main Ideas, Central Themes & Topic Sentences 5.2 Extracting Supporting Details & Factual Information 5.3 Drawing Logical Inferences & Author's Purpose 5.4 Analyzing Text Structure & Drawing Conclusions
6 Chapter 6: Mathematics Knowledge (MK)
6.1 Pre-Algebra & Number Operations: Exponents, Radicals & Order of Operations 6.2 High School Algebra: Linear Equations, Inequalities & Polynomials 6.3 Geometry Fundamentals: Angles, Triangles, Quadrilaterals & Circles 6.4 Coordinate Geometry, Perimeter, Area & Volume Formulas
7 Chapter 7: Electronics Information (EI)
7.1 Electrical Principles: Voltage, Current, Resistance & Ohm's Law (V=IR, P=VI) 7.2 Electric Circuits: Series, Parallel & Combination Circuit Analysis 7.3 Electrical Components: Resistors, Capacitors, Inductors & Transformers 7.4 Electronic Semiconductors, Transistors, Diodes & Logic Gates
8 Chapter 8: Auto & Shop Information (AS)
8.1 Automotive Systems: Internal Combustion Engines, Fuel & Ignition Systems 8.2 Automotive Mechanical Systems: Transmission, Suspension, Brakes & Cooling 8.3 Hand & Power Tools: Identification, Uses & Fasteners 8.4 Shop Practice, Woodworking, Metalworking & Safety Rules
9 Chapter 9: Mechanical Comprehension (MC)
9.1 Simple Machines: Levers, Pulleys, Inclined Planes & Mechanical Advantage 9.2 Force, Motion, Work & Power: Newton's Laws & Energy Transformations 9.3 Fluid Dynamics & Pressure: Pascal's Principle, Hydraulics & Pneumatics 9.4 Mechanical Components: Gears, Belts, Springs & Friction
10 Chapter 10: Spatial Assembly Practice (AO Skill)
10.1 Spatial Connections and Assembly Patterns
6.2 High School Algebra: Linear Equations, Inequalities & Polynomials
Key Takeaways Explain This Topic AI
Solve multi-step linear equations and literal equations by isolating the target variable through inverse operations.
Solve systems of linear equations using substitution or elimination, identifying single solutions, parallel lines (no solution), or identical lines (infinite solutions).
Flip the inequality sign when multiplying or dividing an inequality by a negative number.
Factor polynomials using GCF, difference of squares, trinomial grouping, and apply the Quadratic Formula when factoring is not feasible.
Analyze quadratic functions using the discriminant D = b^2 - 4ac to determine the number and nature of real roots.
Last updated: July 2026
6.2 High School Algebra: Linear Equations, Inequalities & Polynomials
Algebraic reasoning forms the core of high school mathematics and constitutes a major portion of the Mathematics Knowledge skill area. Mastering linear equations, inequalities, polynomial manipulations, and quadratic solutions provides the essential toolkit for solving complex mathematical word problems.
Single-Variable & Multi-Step Linear Equations
A linear equation in one variable is an equation that can be written in the form a x + b = 0 ax + b = 0 a x + b = 0 , where a ≠ 0 a \neq 0 a = 0 . The objective when solving linear equations is to isolate the variable on one side of the equal sign.
Systematic 4-Step Solution Method
Clear Parentheses: Apply the distributive property a ( b + c ) = a b + a c a(b + c) = ab + ac a ( b + c ) = ab + a c .
Clear Fractions: Multiply every term on both sides by the least common denominator (LCD).
Combine Like Terms: Group variable terms on one side and constant terms on the opposite side.
Isolate Variable: Divide or multiply by the variable's coefficient.
Literal Equations (Formula Rearrangement)
Practice questions may ask you to solve a geometry or physics formula for a specific variable.
Example: Solve the trapezoid area formula A = 1 2 h ( b 1 + b 2 ) A = \frac{1}{2}h(b_1 + b_2) A = 2 1 h ( b 1 + b 2 ) for b 1 b_1 b 1 .
Multiply both sides by 2 2 2 : 2 A = h ( b 1 + b 2 ) 2A = h(b_1 + b_2) 2 A = h ( b 1 + b 2 ) .
Divide by h h h : 2 A h = b 1 + b 2 \frac{2A}{h} = b_1 + b_2 h 2 A = b 1 + b 2 .
Subtract b 2 b_2 b 2 : b 1 = 2 A h − b 2 b_1 = \frac{2A}{h} - b_2 b 1 = h 2 A − b 2 .
Systems of Linear Equations
A system of linear equations consists of two or more equations with two or more variables. The solution is the ordered pair ( x , y ) (x, y) ( x , y ) that satisfies both equations simultaneously.
Solution Methods
Substitution Method: Solves one equation for one variable and substitutes that expression into the second equation. Best when one coefficient is 1 1 1 or − 1 -1 − 1 .
Elimination (Addition) Method: Multiplies one or both equations by constants so that adding the equations eliminates one variable. Best for general equations.
Types of Linear Systems
System Classification Graphical Representation Number of Solutions Algebraic Condition
Consistent Independent Intersecting lines Exactly One Solution Slopes are different ( m 1 ≠ m 2 m_1 \neq m_2 m 1 = m 2 )
Inconsistent Parallel lines No Solution Same slope, different intercepts ( m 1 = m 2 , b 1 ≠ b 2 m_1 = m_2, b_1 \neq b_2 m 1 = m 2 , b 1 = b 2 )
Consistent Dependent Coincident (same) line Infinitely Many Solutions Same slope, same intercept ( m 1 = m 2 , b 1 = b 2 m_1 = m_2, b_1 = b_2 m 1 = m 2 , b 1 = b 2 )
Worked Example: Elimination Method
Solve the system: { 3 x + 4 y = 18 — (Eq. 1) 2 x − 3 y = − 5 — (Eq. 2) \begin{cases} 3x + 4y = 18 \quad \text{--- (Eq. 1)} \\ 2x - 3y = -5 \quad \text{--- (Eq. 2)} \end{cases} { 3 x + 4 y = 18 — (Eq. 1) 2 x − 3 y = − 5 — (Eq. 2)
Multiply Eq. 1 by 3 3 3 : 9 x + 12 y = 54 9x + 12y = 54 9 x + 12 y = 54 .
Multiply Eq. 2 by 4 4 4 : 8 x − 12 y = − 20 8x - 12y = -20 8 x − 12 y = − 20 .
Add the equations to eliminate y y y : ( 9 x + 8 x ) + ( 12 y − 12 y ) = 54 − 20 ⟹ 17 x = 34 ⟹ x = 2 (9x + 8x) + (12y - 12y) = 54 - 20 \implies 17x = 34 \implies x = 2 ( 9 x + 8 x ) + ( 12 y − 12 y ) = 54 − 20 ⟹ 17 x = 34 ⟹ x = 2
Substitute x = 2 x = 2 x = 2 into Eq. 1: 3 ( 2 ) + 4 y = 18 ⟹ 6 + 4 y = 18 ⟹ 4 y = 12 ⟹ y = 3 3(2) + 4y = 18 \implies 6 + 4y = 18 \implies 4y = 12 \implies y = 3 3 ( 2 ) + 4 y = 18 ⟹ 6 + 4 y = 18 ⟹ 4 y = 12 ⟹ y = 3
Solution: ( 2 , 3 ) (2, 3) ( 2 , 3 ) .
Linear Inequalities & Absolute Value
Linear inequalities use order symbols: < , ≤ , > , ≥ <, \le, >, \ge < , ≤ , > , ≥ . Solving inequalities follows the same rules as equations, with one vital exception.
GOLDEN RULE OF INEQUALITIES: When multiplying or dividing both sides of an inequality by a negative number , you MUST reverse the inequality direction ( < < < becomes > > > , ≤ \le ≤ becomes ≥ \ge ≥ , etc.).
Example: Negative Division − 4 x + 7 ≥ 27 ⟹ − 4 x ≥ 20 ⟹ − 4 x − 4 ≤ 20 − 4 ⟹ x ≤ − 5 -4x + 7 \ge 27 \implies -4x \ge 20 \implies \frac{-4x}{-4} \le \frac{20}{-4} \implies x \le -5 − 4 x + 7 ≥ 27 ⟹ − 4 x ≥ 20 ⟹ − 4 − 4 x ≤ − 4 20 ⟹ x ≤ − 5
Absolute Value Equations & Inequalities
Absolute value ∣ x ∣ |x| ∣ x ∣ represents distance from zero on a number line (always non-negative).
Equation: ∣ x ∣ = k ⟹ x = k or x = − k ( k ≥ 0 ) |x| = k \implies x = k \quad \text{or} \quad x = -k \quad (k \ge 0) ∣ x ∣ = k ⟹ x = k or x = − k ( k ≥ 0 ) .
"Less Than" Inequality (AND): ∣ x ∣ < k ⟹ − k < x < k |x| < k \implies -k < x < k ∣ x ∣ < k ⟹ − k < x < k .
"Greater Than" Inequality (OR): ∣ x ∣ > k ⟹ x < − k or x > k |x| > k \implies x < -k \quad \text{or} \quad x > k ∣ x ∣ > k ⟹ x < − k or x > k .
Polynomial Operations & Factoring
A polynomial is an expression consisting of variables and coefficients combined using addition, subtraction, and non-negative integer exponents.
Polynomial Operations
Multiplying Binomials (FOIL): ( a + b ) ( c + d ) = First ( a c ) + Outer ( a d ) + Inner ( b c ) + Last ( b d ) (a + b)(c + d) = \text{First } (ac) + \text{Outer } (ad) + \text{Inner } (bc) + \text{Last } (bd) ( a + b ) ( c + d ) = First ( a c ) + Outer ( a d ) + Inner ( b c ) + Last ( b d ) . ( 2 x − 5 ) ( 3 x + 4 ) = 6 x 2 + 8 x − 15 x − 20 = 6 x 2 − 7 x − 20 (2x - 5)(3x + 4) = 6x^2 + 8x - 15x - 20 = 6x^2 - 7x - 20 ( 2 x − 5 ) ( 3 x + 4 ) = 6 x 2 + 8 x − 15 x − 20 = 6 x 2 − 7 x − 20
Core Factoring Techniques
Greatest Common Factor (GCF): Factor out the highest common factor from all terms. 12 x 3 − 18 x 2 = 6 x 2 ( 2 x − 3 ) 12x^3 - 18x^2 = 6x^2(2x - 3) 12 x 3 − 18 x 2 = 6 x 2 ( 2 x − 3 )
Difference of Two Squares: a 2 − b 2 = ( a − b ) ( a + b ) a^2 - b^2 = (a - b)(a + b) a 2 − b 2 = ( a − b ) ( a + b ) . 16 x 2 − 49 = ( 4 x − 7 ) ( 4 x + 7 ) 16x^2 - 49 = (4x - 7)(4x + 7) 16 x 2 − 49 = ( 4 x − 7 ) ( 4 x + 7 )
Perfect Square Trinomials: a 2 ± 2 a b + b 2 = ( a ± b ) 2 a^2 \pm 2ab + b^2 = (a \pm b)^2 a 2 ± 2 ab + b 2 = ( a ± b ) 2 . x 2 − 10 x + 25 = ( x − 5 ) 2 x^2 - 10x + 25 = (x - 5)^2 x 2 − 10 x + 25 = ( x − 5 ) 2
Trinomial Factoring ( x 2 + b x + c x^2 + bx + c x 2 + b x + c ): Find two numbers that multiply to c c c and add to b b b . x 2 − 7 x + 12 = ( x − 3 ) ( x − 4 ) (since − 3 × − 4 = 12 and − 3 + − 4 = − 7 x^2 - 7x + 12 = (x - 3)(x - 4) \quad \text{(since } -3 \times -4 = 12 \text{ and } -3 + -4 = -7 x 2 − 7 x + 12 = ( x − 3 ) ( x − 4 ) (since − 3 × − 4 = 12 and − 3 + − 4 = − 7
Quadratic Equations & The Quadratic Formula
A quadratic equation is a second-degree polynomial equation: a x 2 + b x + c = 0 ( a ≠ 0 ) ax^2 + bx + c = 0 \quad (a \neq 0) a x 2 + b x + c = 0 ( a = 0 ) .
The Quadratic Formula
For any quadratic equation a x 2 + b x + c = 0 ax^2 + bx + c = 0 a x 2 + b x + c = 0 , the roots are given by: x = − b ± b 2 − 4 a c 2 a x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} x = 2 a − b ± b 2 − 4 a c
The Discriminant ( D = b 2 − 4 a c D = b^2 - 4ac D = b 2 − 4 a c )
The expression under the square root, D = b 2 − 4 a c D = b^2 - 4ac D = b 2 − 4 a c , reveals the number and nature of roots:
If D > 0 D > 0 D > 0 and a perfect square: Two distinct rational real roots.
If D > 0 D > 0 D > 0 and not a perfect square: Two distinct irrational real roots.
If D = 0 D = 0 D = 0 : Exactly one repeated rational real root ( x = − b / 2 a x = -b / 2a x = − b /2 a ).
If D < 0 D < 0 D < 0 : No real roots (two complex conjugate roots).
Worked Example: Quadratic Solution
Solve 2 x 2 − 5 x − 3 = 0 2x^2 - 5x - 3 = 0 2 x 2 − 5 x − 3 = 0 .
Identify coefficients: a = 2 a = 2 a = 2 , b = − 5 b = -5 b = − 5 , c = − 3 c = -3 c = − 3 .
Calculate discriminant: D = ( − 5 ) 2 − 4 ( 2 ) ( − 3 ) = 25 + 24 = 49 D = (-5)^2 - 4(2)(-3) = 25 + 24 = 49 D = ( − 5 ) 2 − 4 ( 2 ) ( − 3 ) = 25 + 24 = 49 .
Apply Quadratic Formula: x = − ( − 5 ) ± 49 2 ( 2 ) = 5 ± 7 4 x = \frac{-(-5) \pm \sqrt{49}}{2(2)} = \frac{5 \pm 7}{4} x = 2 ( 2 ) − ( − 5 ) ± 49 = 4 5 ± 7
Solution 1: x = 5 + 7 4 = 12 4 = 3 x = \frac{5 + 7}{4} = \frac{12}{4} = 3 x = 4 5 + 7 = 4 12 = 3 .
Solution 2: x = 5 − 7 4 = − 2 4 = − 1 2 x = \frac{5 - 7}{4} = \frac{-2}{4} = -\frac{1}{2} x = 4 5 − 7 = 4 − 2 = − 2 1 .
Solution set: { − 1 2 , 3 } \{ -\frac{1}{2}, 3 \} { − 2 1 , 3 } .