Questions
3 questions per exam
Difficulty
Medium
Importance
Fundamental for Linear Programming problems
Overview
Linear inequalities involve relationships expressed by symbols like <, >, ≤, or ≥, forming the backbone of linear programming and optimization problems. Mastering these is essential for board exams as they translate real-world constraints into solvable mathematical models. Students must focus on manipulating algebraic expressions and accurately visualizing solution regions on the Cartesian plane.
Algebraic Solutions for Single Variable
Solving single-variable linear inequalities follows algebraic rules similar to equations, but with one critical distinction regarding signs. When multiplying or dividing both sides by a negative number, the inequality sign must be reversed.
- Add or subtract the same value from both sides without changing the sign
- Reverse the inequality operator when multiplying/dividing by a negative real number
- Solution sets are expressed using interval notation like [a, b) or (-∞, a]
- Verify solutions by testing a value from the derived interval
Graphical Solutions on 2D Plane
Graphical representation involves plotting the corresponding linear equation as a boundary line and determining the correct half-plane. A solid line is used for inclusive inequalities (≤ or ≥), while a dashed line represents strict inequalities (< or >).
- Convert the inequality to an equality (y = mx + c) to plot the boundary
- Use a dashed line for strict inequalities and solid for inclusive
- Select the origin (0,0) as a test point to determine the shaded region
- If the test point satisfies the inequality, shade the side containing it
Systems of Linear Inequalities
A system of inequalities requires identifying the common region where all individual solution sets overlap. This feasible region represents the set of all points that satisfy every constraint simultaneously.
- Shade regions for each individual inequality separately
- The solution is the intersection area shared by all shaded regions
- Check for unbounded regions that extend to infinity
- Ensure all boundary line intersections are correctly calculated as vertices
Formula Sheet
ax + b < 0
ax + by ≤ c
Reversal Rule: If a < b and c < 0, then ac > bc
Exam Tip
Always test the point (0,0) first, unless the line passes through the origin, in which case use (1,0) or (0,1) to definitively identify the shaded region.
Common Mistakes
- Failing to flip the inequality sign when dividing by a negative number.
- Using a solid line instead of a dashed line for strict inequalities like < or >.
- Misidentifying the feasible region by testing a point that actually lies on the boundary line.
More Revision Notes
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