Section 5.3: Linear Inequalities and Systems of Linear Inequalities

Section 5.3: Linear Inequalities

Introduction to Linear Inequalities

Linear inequalities extend the concept of linear equations by introducing inequality symbols (<, >, ≤, ≥) instead of the equality sign. They are used to describe regions of the coordinate plane that satisfy certain conditions, which is especially useful in business and optimization problems.

• Linear Equation: An equation of the form $y = mx + b$ represents a straight line.

• Linear Inequality: An inequality such as $y < mx + b$ or $y \geq mx + b$ describes a half-plane, either above or below the line.

• Solution Set: The set of all points $(x, y)$ that satisfy the inequality.

Graphing Linear Inequalities

To graph a linear inequality, follow these steps:

1. If necessary, rewrite the inequality in slope-intercept form ($y = mx + b$).

2. Graph the corresponding linear equation:

   • Use a dashed line for strict inequalities (< or >).

   • Use a solid line for inequalities that include equality (≤ or ≥).

3. Choose a test point not on the line (often (0,0) if possible).

4. Substitute the test point into the inequality:

   • If true, shade the half-plane containing the test point.

   • If false, shade the opposite half-plane.

Example 1: Graphing $y < 2x + 1$

• Graph the line $y = 2x + 1$ as a dashed line.

• Test point (3,0): $0 < 2(3) + 1 \implies 0 < 7$ (true), so shade the side containing (3,0).

Example 2: Graphing $2x - y < 4$

• Rewrite as $y > 2x - 4$ (or $y \leq -\frac{1}{2}x + 2$ after manipulation).

• Graph the line $y = -\frac{1}{2}x + 2$ as a solid line (since the inequality is ≤).

• Test point (0,0): $0 \leq -\frac{1}{2}(0) + 2 \implies 0 \leq 2$ (true), so shade the side containing (0,0).

Example 3: Application – Nut Mix Cost

A store wants to create a nut mix with peanuts ($4/lb$) and cashews ($8/lb$) costing $6/lb$ or less. Let $p$ = pounds of peanuts, $c$ = pounds of cashews in 1 lb of mix. The inequality is $4p + 8c \leq 6$.

• Find intercepts:

  • If $p = 0$: $8c = 6 \implies c = 0.75$

  • If $c = 0$: $4p = 6 \implies p = 1.5$

• Graph the line and shade the region below it in the first quadrant.

Systems of Linear Inequalities

Introduction to Systems

A system of linear inequalities consists of two or more inequalities. The solution set is the region where all shaded areas overlap, representing all points that satisfy every inequality in the system.

• Graphical Solution: Graph each inequality and identify the intersection (overlap) of all solution regions.

Example 4: Graphing a System

Graph the solution to the system:

• $y \leq 2x + 1$

• $y \geq x - 1$

• Graph each inequality and shade the appropriate region.

• The solution is the region where the shaded areas overlap.

Example 5: Business Application – Production Constraints

A company produces basic and premium products. Each basic product requires 20 minutes of assembly and 15 minutes of painting; each premium product requires 30 minutes of assembly and 30 minutes of painting. The company has 3,900 minutes of assembly and 3,300 minutes of painting available per week. Let $b$ = number of basic products, $p$ = number of premium products.

• Assembly constraint: $20b + 30p \leq 3900$

• Painting constraint: $15b + 30p \leq 3300$

• Non-negativity: $b \geq 0$, $p \geq 0$

• Graph the inequalities; the feasible region shows all possible combinations of products that can be produced within staffing limits.

Applications and Further Context

• These techniques are foundational for linear programming, a branch of mathematics used for optimization in business and economics.

• By analyzing the feasible region, companies can determine the best production mix to maximize profit or minimize cost.

Summary Table: Key Steps for Graphing Linear Inequalities

Step	Description
1	Rewrite the inequality in slope-intercept form if necessary.
2	Graph the boundary line (dashed for < or >, solid for ≤ or ≥).
3	Choose a test point not on the line.
4	Substitute the test point into the inequality.
5	Shade the region where the inequality is true.

Additional info: Linear inequalities and their systems are essential for modeling real-world constraints and making optimal decisions in various fields, including business, economics, and health sciences.