Linear Inequalities

Inequality Fundamentals and Intervals

Inequalities describe relationships where values are not necessarily equal, mapped visually onto a number line using strict inequality and non-strict inequality notation.

$$x \in (a, b) \implies a < x < b$$

Interval notation converts inequality ranges into concise set representations.

$($ or $)$=Exclusive boundary (strict)(unitless)

$[$ or $]$=Inclusive boundary (non-strict)(unitless)

$x < a$

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Uses parenthesis $(-\infty, a)$

$x \ge a$

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Uses bracket $[a, \infty)$

Visual Cues: Open circles (strict) indicate the boundary is not part of the solution set.

Bracket Logic: Parentheses correlate to $<$ and $>$, while square brackets correlate to $\le$ and $\ge$.

The Negative Sign Reversal Rule

Unlike equations, multiplying or dividing an inequality by a negative value reverses the order of the relationship to maintain truth.

$$a < b \text{ and } c < 0 \implies ac > bc$$

Scaling by a negative constant flips the direction of the inequality sign.

$a, b$=Initial values(real numbers)

$c$=Negative multiplier($c < 0$)

$c = -1$

→

$x < 5 \implies -x > -5$

Sign Reversal: This occurs because negative scaling reflects points across the origin, swapping their relative order.

Additive Invariance: Adding or subtracting any value does NOT change the inequality sign.

Solving Systems and Multi-Step Logic

Solving multi-step inequalities follows a hierarchy: simplify terms, isolating variables, and applying the multiplicative property with caution.

$$ax + b < cx + d \implies (a-c)x < d - b$$

Standard algebraic isolation techniques apply until the final coefficient division.

$a, c$=Coefficients of $x$(unitless)

$b, d$=Constant terms(unitless)

$a-c < 0$

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Final step requires sign flip

LCM Technique: When dealing with fractions, multiply the entire inequality by the least common multiple to simplify.

Compound Inequalities: Solutions for expressions like $a < f(x) < b$ represent the intersection of two conditions.

Two-Variable Inequalities and Regions

Linear inequalities in two variables define a half-plane in the coordinate system, bounded by a boundary line and verified via a test point.

$$ax + by \le c$$

The linear equation $ax + by = c$ creates the boundary of the solution region.

$ax+by=c$=Boundary line (associated equation)(linear)

$<$ or $>$=Dashed boundary (points on line excluded)(exclusive)

$\le$ or $\ge$=Solid boundary (points on line included)(inclusive)

Testing (0,0)

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Simplest test point when the line doesn't pass through the origin

Line passes through origin

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Use any other convenient point like $(1, 0)$ or $(0, 1)$ as the test point

Graphing Procedure: (1) Graph the associated equation $ax + by = c$, using dashes for strict and solid for non-strict. (2) Pick a test point and substitute — shade the half-plane that satisfies the inequality.

Vertical Lines: $x \ge a$ shades the right half-plane; $x \le a$ shades the left half-plane.

Horizontal Lines: $y \ge b$ shades the upper half-plane; $y \le b$ shades the lower half-plane.

Systems of Inequalities and Solution Regions

The solution of a system of inequalities is the intersection of all individual half-plane regions, forming a common area called the solution region.

Intersection Method: Graph each inequality separately on the same axes, then identify the overlapping (common) region.

Corner Points: Where two boundary lines of the solution region intersect, they form a corner point (vertex). These are found by solving pairs of associated equations simultaneously.

Bounded vs Unbounded: A solution region that can be enclosed in a circle is bounded; one extending infinitely is unbounded.

Convex Property: All feasible regions from linear inequalities are convex — any line segment joining two points within the region lies entirely inside it.

Feasible Region and Constraints

When a system of inequalities includes non-negative constraints ($x \ge 0$, $y \ge 0$), the solution region is restricted to the first quadrant and is called the feasible region.

Problem Constraints: The inequalities that model real-world limitations (resources, capacity, budgets) on decision variables.

Non-Negative Constraints: $x \ge 0$ and $y \ge 0$ restrict the solution to the first quadrant — quantities like production count or hours cannot be negative.

Feasible Solution: Any point $(x, y)$ inside or on the boundary of the feasible region that satisfies all constraints simultaneously.

Finding Corner Points: Solve each pair of boundary equations simultaneously. The intersection points that lie within the feasible region are the corner points.

Linear Programming and Optimization

Linear programming is the method of maximizing or minimizing an objective function subject to a system of linear inequality constraints, with the solution always occurring at a corner point of the feasible region.

$$f(x, y) = ax + by$$

The objective function is a linear expression whose value we want to optimize.

$f(x, y)$=Objective function (profit, cost, etc.)(depends on context)

$a, b$=Coefficients representing per-unit contribution(Rs/unit, etc.)

$x, y$=Decision variables (quantities to determine)(units of product/resource)

Feasible region is unbounded

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Maximum may not exist (value can grow without bound), but minimum may still exist

Optimal value at two adjacent corner points

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Every point on the line segment between them also gives the same optimal value

Corner Point Theorem: The maximum and minimum values of a linear objective function over a convex feasible region always occur at one or more corner points (vertices).

Optimization Procedure: (1) Graph the feasible region from all constraints. (2) Identify all corner points. (3) Evaluate $f(x,y)$ at each corner point. (4) The largest value is the maximum; the smallest is the minimum.

Why Only Corners?: A linear function on a convex polygon increases steadily in one direction — it cannot have an interior peak or valley, so extrema must lie on the boundary, specifically at vertices.

Converting a word problem into a linear programming model requires identifying decision variables, writing problem constraint inequalities, and defining the objective function.

Step 1 — Variables: Define $x$ and $y$ as the quantities to be determined (e.g., units of product A and B).

Step 2 — Constraints: Translate each resource limitation into an inequality (e.g., $5x + 4y \le 120$ for machine hours).

Step 3 — Non-Negativity: Always include $x \ge 0$, $y \ge 0$ since physical quantities cannot be negative.

Step 4 — Objective: Write the function to optimize (e.g., Profit $= 40x + 50y$, or Cost $= 25x + 30y$).