ExamFinch.com My Study

Study guide

6.3 Linear Equations, Inequalities & Absolute Value

Practice this exam

Armed Forces Qualification Test AFQT

Track your progress as you read

Finished sections get a check mark.

1 Chapter 1: AFQT Structure, Scoring & Enlistment Requirements

1.1 AFQT Overview, Subtest Breakdown & CAT-ASVAB vs. P&P Format 1.2 The AFQT Scoring Formula (2×VE + AR + MK) & Percentile Norming 1.3 Branch Minimum Scores, AFQT Categories & Retake Policies

2 Chapter 2: Arithmetic Reasoning (AR) — Ratios, Rates & Percentages

2.1 Ratios, Proportions & Scale Conversions 2.2 Percentages, Discounts, Interest & Tax Calculations 2.3 Rate, Distance, Time & Speed Word Problems 2.4 Work Rates & Combined Labor Problems

3 Chapter 3: Arithmetic Reasoning (AR) — Mixtures, Sequences & Applied Math

3.1 Mixture, Concentration & Price-Blending Problems 3.2 Sequences, Patterns & Series Calculations 3.3 Averages, Weighted Averages & Statistical Word Problems

4 Chapter 4: Word Knowledge (WK) — Vocabulary & Word Structure

4.1 High-Frequency AFQT Vocabulary & Synonym Recognition 4.2 Latin & Greek Roots, Prefixes & Suffixes for Word Decoding 4.3 Context Clues & Words with Multiple Meanings

5 Chapter 5: Paragraph Comprehension (PC) — Reading Analysis & Inference

5.1 Identifying Main Ideas & Primary Purpose 5.2 Locating Supporting Details & Direct Facts 5.3 Drawing Logical Inferences & Implied Conclusions 5.4 Analyzing Tone, Author Perspective & Contextual Vocabulary

6 Chapter 6: Mathematics Knowledge (MK) — Fundamentals & Linear Algebra

6.1 Arithmetic Operations, Order of Operations & Number Properties 6.2 Exponents, Scientific Notation & Square Roots / Radicals 6.3 Linear Equations, Inequalities & Absolute Value 6.4 Systems of Linear Equations & Word-to-Algebra Translation

7 Chapter 7: Mathematics Knowledge (MK) — Advanced Algebra & Functions

7.1 Polynomial Operations & Factoring Techniques 7.2 Quadratic Equations, Factoring & Quadratic Formula 7.3 Coordinate Geometry: Slope, Distance, Midpoint & Linear Graphs

8 Chapter 8: Mathematics Knowledge (MK) — Geometry & Spatial Math

8.1 Angles, Triangles & Pythagorean Theorem 8.2 Quadrilaterals, Polygons & Circle Geometry 8.3 3D Geometry: Surface Area & Volume of Solids 8.4 Basic Geometric Transformations & Trigonometric Ratios

6.3 Linear Equations, Inequalities & Absolute Value

Key Takeaways Explain This Topic AI

Solving linear equations requires maintaining equality through inverse operations to isolate the unknown variable on one side.

When solving linear inequalities, multiplying or dividing both sides by a negative number reverses the direction of the inequality sign (e.g., -2x < 8 becomes x > -4).

Absolute value equations of the form |x| = a (where a >= 0) split into two distinct linear equations: x = a or x = -a.

Absolute value inequalities split into compound expressions: |x| < a translates into a conjunction (-a < x < a), whereas |x| > a translates into a disjunction (x < -a or x > a).

Last updated: July 2026

6.3 Linear Equations, Inequalities & Absolute Value

Linear equations and inequalities represent fundamental mathematical relationships where variables are raised exclusively to the first power. The AFQT Mathematics Knowledge subtest requires rapid, accurate execution of algebraic techniques to isolate variables, handle inequality flips, and solve absolute value conditions.

Solving Single-Variable Linear Equations

A linear equation in one variable can be written in the standard form a x + b = c ax + b = c a x + b = c , where a ≠ 0 a \neq 0 a  = 0 . Isolating x x x requires applying properties of equality (doing the exact same mathematical operation to both sides of the equation).

Systematic 5-Step Equation Solver Framework

Clear Fractions and Decimals : Multiply every term by the Least Common Denominator (LCD) of all fractions or by an appropriate power of 10 for decimals.

Simplify Both Sides Independently : Expand parentheses using the Distributive Property ( a ( b + c ) = a b + a c a(b+c) = ab + ac a ( b + c ) = ab + a c ) and combine like terms on each side.

Isolate Variable Terms : Use addition or subtraction properties of equality to collect all terms containing the variable on one side.

Isolate Constant Terms : Use addition or subtraction properties of equality to move all numerical constants to the opposite side.

Isolate the Variable : Multiply or divide by the variable's coefficient to achieve x = value x = \text{value} x = value , then check by back-substituting into the original equation.

Comprehensive Multi-Step Example

Solve for x x x : 3 ( x − 2 ) 4 − x + 1 3 = 1 2 \frac{3(x - 2)}{4} - \frac{x + 1}{3} = \frac{1}{2} 4 3 ( x − 2 ) ​ − 3 x + 1 ​ = 2 1 ​

Clear Fractions : The denominators are 4, 3, and 2. The LCD is 12. Multiply every term on both sides by 12: 12 ⋅ [ 3 ( x − 2 ) 4 ] − 12 ⋅ [ x + 1 3 ] = 12 ⋅ [ 1 2 ] 12 \cdot \left[\frac{3(x - 2)}{4}\right] - 12 \cdot \left[\frac{x + 1}{3}\right] = 12 \cdot \left[\frac{1}{2}\right] 12 ⋅ [ 4 3 ( x − 2 ) ​ ] − 12 ⋅ [ 3 x + 1 ​ ] = 12 ⋅ [ 2 1 ​ ] 3 ⋅ [ 3 ( x − 2 ) ] − 4 ⋅ [ x + 1 ] = 6 3 \cdot [3(x - 2)] - 4 \cdot [x + 1] = 6 3 ⋅ [ 3 ( x − 2 )] − 4 ⋅ [ x + 1 ] = 6 9 ( x − 2 ) − 4 ( x + 1 ) = 6 9(x - 2) - 4(x + 1) = 6 9 ( x − 2 ) − 4 ( x + 1 ) = 6

Distribute & Combine Like Terms : 9 x − 18 − 4 x − 4 = 6 9x - 18 - 4x - 4 = 6 9 x − 18 − 4 x − 4 = 6 5 x − 22 = 6 5x - 22 = 6 5 x − 22 = 6

  • Isolate Constant & Variable : 5 x = 6 + 22 ⟹ 5 x = 28 5x = 6 + 22 \implies 5x = 28 5 x = 6 + 22 ⟹ 5 x = 28
  • Solve for x x x : x = 28 5 or 5.6 x = \frac{28}{5} \quad \text{or} \quad 5.6 x = 5 28 ​ or 5.6
  • Special Solution Outcomes

Single Unique Solution : Occurs when x x x equals a specific numerical value (e.g., x = 4 x = 4 x = 4 ).

No Solution (Inconsistent Equation) : Occurs when variable terms cancel completely, leaving a false statement (e.g., 0 = 7 0 = 7 0 = 7 ).

Infinitely Many Solutions (Identity Equation) : Occurs when variable terms cancel completely, leaving a true statement (e.g., 5 = 5 5 = 5 5 = 5 or 0 = 0 0 = 0 0 = 0 ).

Linear Inequalities in One Variable

Linear inequalities use order symbols ( < , ≤ , > , ≥ <, \le, >, \ge < , ≤ , > , ≥ ) to describe a range of solutions rather than a single fixed point.

Inequality Symbol Meaning Circle on Number Line Included in Interval?

  • < < < Strictly less than Open Circle ( ○ ) No
  • > > > Strictly greater than Open Circle ( ○ ) No
  • ≤ \le ≤ Less than or equal to Closed Solid Circle ( ● ) Yes
  • ≥ \ge ≥ Greater than or equal to Closed Solid Circle ( ● ) Yes

⚡ THE GOLDEN RULE OF INEQUALITIES : Whenever you multiply or divide both sides of an inequality by a negative number , you MUST reverse (flip) the direction of the inequality symbol.

Why Does the Inequality Sign Flip?

Consider the true numerical statement: 3 < 7 3 < 7 3 < 7 .

Multiply both sides by + 2 ⟹ 6 < 14 +2 \implies 6 < 14 + 2 ⟹ 6 < 14 (True).

Multiply both sides by − 2 ⟹ − 6 vs − 14 -2 \implies -6 \text{ vs } -14 − 2 ⟹ − 6 vs − 14 .

On a number line, − 6 -6 − 6 lies to the right of − 14 -14 − 14 , so − 6 > − 14 -6 > -14 − 6 > − 14 . The sign must flip to remain true.

  • Multi-Step Inequality Example
  • Solve for x x x : − 4 ( 2 x − 3 ) ≥ 36 -4(2x - 3) \ge 36 − 4 ( 2 x − 3 ) ≥ 36
  • Distribute − 4 -4 − 4 : − 8 x + 12 ≥ 36 -8x + 12 \ge 36 − 8 x + 12 ≥ 36
  • Subtract 12 from both sides: − 8 x ≥ 24 -8x \ge 24 − 8 x ≥ 24

Divide both sides by − 8 -8 − 8 ( MUST FLIP INEQUALITY SIGN FROM ≥ \ge ≥ TO ≤ \le ≤ ): x ≤ 24 − 8 ⟹ x ≤ − 3 x \le \frac{24}{-8} \implies x \le -3 x ≤ − 8 24 ​ ⟹ x ≤ − 3

Solution Set = All real numbers less than or equal to − 3 -3 − 3 (Interval: ( − ∞ , − 3 ] (-\infty, -3] ( − ∞ , − 3 ] ).

Absolute Value Equations & Inequalities

The absolute value of a real number, written ∣ x ∣ |x| ∣ x ∣ , represents its non-negative distance from zero on the real number line. Because distance is never negative, ∣ x ∣ ≥ 0 |x| \ge 0 ∣ x ∣ ≥ 0 for all real numbers.

Absolute Value Equations: ∣ a x + b ∣ = c |ax + b| = c ∣ a x + b ∣ = c

If c < 0 c < 0 c < 0 : No Solution (distance cannot be negative).

If c = 0 c = 0 c = 0 : Single solution a x + b = 0 ax + b = 0 a x + b = 0 .

If c > 0 c > 0 c > 0 : Split into two separate linear equations : a x + b = c OR a x + b = − c ax + b = c \quad \text{OR} \quad ax + b = -c a x + b = c OR a x + b = − c

Example: Solve ∣ 3 x − 5 ∣ = 16 |3x - 5| = 16 ∣3 x − 5∣ = 16 3 x − 5 = 16 OR 3 x − 5 = − 16 3 x = 21 3 x = − 11 x = 7 x = − 11 3 \begin{array}{rccl}

  • 3x - 5 = 16 & \text{OR} & 3x - 5 = -16 \\
  • 3x = 21 & & 3x = -11 \\
  • x = 7 & & x = -\frac{11}{3}
  • \end{array} 3 x − 5 = 16 3 x = 21 x = 7 ​ OR ​ 3 x − 5 = − 16 3 x = − 11 x = − 3 11 ​ ​

Both solutions must be checked in the original absolute value equation.

Absolute Value Inequalities

To solve absolute value inequalities where c > 0 c > 0 c > 0 , memorize the split rules:

Less Than / Less Than or Equal To ("Less thAND") : ∣ a x + b ∣ ≤ c |ax + b| \le c ∣ a x + b ∣ ≤ c Translates into a three-part continuous compound conjunction : − c ≤ a x + b ≤ c -c \le ax + b \le c − c ≤ a x + b ≤ c

Example : Solve ∣ 2 x − 7 ∣ < 9 |2x - 7| < 9 ∣2 x − 7∣ < 9 − 9 < 2 x − 7 < 9 ⟹ − 2 < 2 x < 16 ⟹ − 1 < x < 8 -9 < 2x - 7 < 9 \implies -2 < 2x < 16 \implies -1 < x < 8 − 9 < 2 x − 7 < 9 ⟹ − 2 < 2 x < 16 ⟹ − 1 < x < 8

Greater Than / Greater Than or Equal To ("GreatOR") : ∣ a x + b ∣ ≥ c |ax + b| \ge c ∣ a x + b ∣ ≥ c Translates into a two-part disjunction joined by OR : a x + b ≤ − c OR a x + b ≥ c ax + b \le -c \quad \text{OR} \quad ax + b \ge c a x + b ≤ − c OR a x + b ≥ c

Example : Solve ∣ 4 x + 3 ∣ ≥ 15 |4x + 3| \ge 15 ∣4 x + 3∣ ≥ 15 4 x + 3 ≤ − 15 OR 4 x + 3 ≥ 15 4x + 3 \le -15 \quad \text{OR} \quad 4x + 3 \ge 15 4 x + 3 ≤ − 15 OR 4 x + 3 ≥ 15 4 x ≤ − 18 ⟹ x ≤ − 4.5 OR 4 x ≥ 12 ⟹ x ≥ 3 4x \le -18 \implies x \le -4.5 \quad \text{OR} \quad 4x \ge 12 \implies x \ge 3 4 x ≤ − 18 ⟹ x ≤ − 4.5 OR 4 x ≥ 12 ⟹ x ≥ 3