Ch 05 - Linear Inequalities11th Mathematics
Strict vs. Slack Inequalities
What is the strict mathematical distinction between 'Strict' and 'Slack' inequalities?
What is the strict mathematical distinction between 'Strict' and 'Slack' inequalities?
- Strict Inequalities: Employ strictly the symbols
<(less than) or>(greater than). The geometric boundary is NOT included in the solution set. - Slack Inequalities: Employ the symbols
\leq(less than or equal to) or\geq(greater than or equal to). The geometric boundary IS strictly included in the solution set.
Prism Memory Hook
Slack cuts you some SLACK (allows equality). Strict is STRICT (no equality).
Number Line Plotting Conventions
What is the standard graphical convention for representing strict vs. slack inequalities on a 1D number line?
What is the standard graphical convention for representing strict vs. slack inequalities on a 1D number line?
- Open Circle (): Used specifically for strict inequalities (
<or>). Indicates the boundary point is mathematically excluded from the domain. - Closed/Dark Circle (): Used specifically for slack inequalities (
\leqor\geq). Indicates the boundary point is mathematically included in the domain.
Prism Memory Hook
Slack is packed (closed circle). Strict is stripped (open circle).
Inequality Rule 1: Addition & Subtraction
Inequality Rule 1: Addition & Subtraction
State the rule for adding or subtracting the same number on both sides of an inequality.
Derivation
Both sides of a linear inequality can have equal numbers added to (or subtracted from) them without changing the orientation of the inequality sign, as the relative distance shifts uniformly.
Component Anatomy:
- : Any real numbers or algebraic expressions mapped on a number line.
Boundary Conditions / Domain:
- Valid universally for all real numbers . The inequality sign is strictly invariant under addition or subtraction.
Prism Memory Hook
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Also in this chapter
27 more cards to study in Prism
- 4Inequality Rule 2a: Multiplication/Division by a Positive
- 5Inequality Rule 2b: Multiplication by a Negative
- 6Domain Trap: Natural vs. Integers vs. Real Numbers
- 7Replacement Set vs. Solution Set
- 8SOP: Solving Double Inequalities
- 9The Cartesian Half-Planes
- 10Solid vs. Dotted Boundary Lines (2D Graphing)
- 11SOP: Test Point Method (Part 1 - Boundary Line)
- 12Feasible Region in Systems of Inequalities
- 13Bounded vs. Unbounded Solution Regions
- 14First Quadrant Confinement Restrictions
- 15The 'No Solution' / Empty Feasible Region
- 16SOP: Reverse Engineering Graph (Part 1 - Equation)
- 17Modulus Inequality: Less Than (Bounded)
- 18Modulus Inequality: Greater Than (Unbounded Split)
- 19Shifted Modulus Inequality Boundary Mapping
- 20Denominator Non-Zero Constraint in Rational Inequalities
- 21SOP: Complex Rational Modulus (Part 1 - Setup)
- 22SOP: Unknown Denominator Sign Trap
- 23Word Problem Translation Trap: At Least vs. At Most
- 24SOP: Consecutive Odd/Even Integer Problems
- 25SOP: Average Score Constraints
- 26SOP: Geometric Constraints (Part 1 - Setup)
- 27SOP: Acid Dilution (Part 1 - Mass Balance)
- 28Manufacturing Profit Constraints
- 29Intelligence Quotient (IQ) Range Modeling
- 30Temperature Conversion Inequality
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