Equations, Inequalities, and Functions: College Algebra Study Notes

Equations, Inequalities, and Functions

Introduction

This study guide covers foundational topics in College Algebra, including equations, inequalities, systems of equations, linear programming, and functions. Each section provides definitions, properties, solution methods, and practical examples to support your understanding and exam preparation.

Equations

Definition of an Equation

• Equation: A statement that two algebraic expressions are equal.

• Examples:

  • $2x = 4$

  • $x^2 = 3x + 7$

  • $x^3 - 3x^2 + 4x - 1 = 0$

• Common types: Linear equations, quadratic equations, polynomial equations.

Linear Equations

• Definition: Equations where variables have degree one (no exponents other than 1).

• General form: $y = mx + c$, where $m$ is the slope and $c$ is the y-intercept.

• Examples:

  • $x + 4 = 6$

  • $y + 3x = 7$

  • $ax + b = y$

Linearity Criteria

• All variables have exponent 1.

• No products or quotients of variables.

Example: Identifying Linear Equations

Equation	Linear?	Reason
$x + \sqrt{3}y = 7$	Yes	All variables degree 1
$y + xy = 5$	No	Product $xy$
$x = -7y + 2$	Yes	All variables degree 1
$kx - \frac{1}{2}x^2 + k = 0$	No	$x^2$ term
$2x + 7y - x = 1$	Yes	All variables degree 1
$2x - \frac{y}{x} + 7 = 0$	No	Quotient $\frac{y}{x}$

Solving Linear Equations

• Isolate the variable to one side of the equation.

• Example: Solve $x + 4 = 0$ $x = -4$

• Example: Solve $\frac{2}{5}x = 6$ $x = 15$

Word Problem Example

• Three parcels X, Y, Z have total mass 168 kg. X is 20 kg less than Z, Y is twice the total mass of X and Z. Find each mass.

• Solution: Let $X = t$, $Z = t + 20$, $Y = 2(t + t + 20) = 4t + 40$ $t + (t + 20) + (4t + 40) = 168 \implies t = 18$ Masses: X = 18 kg, Y = 112 kg, Z = 38 kg

Systems of Linear Equations

Definition and Types

• A system of linear equations consists of two or more linear equations with the same variables.

• General form: $c_1x + c_2y = d$, $c_3x + c_4y = b$

Possible Solutions

1. Exactly one solution (lines intersect at one point)

2. No solution (lines are parallel)

3. Infinitely many solutions (lines coincide)

Methods of Solution

• Elimination method

• Substitution method

• Graphical method

• Matrix method

• Calculator or computational tools

Elimination Method Example

• Solve:

  • $x - y = 4$

  • $2x - 3y = 11$

• Multiply first equation by 2 and subtract from second: $2x - 2y = 8$ $2x - 3y = 11$ Subtract: $y = -3$ Substitute back: $x = 1$ Solution: $(x, y) = (1, -3)$

Substitution Method Example

• Solve:

  • $x + y = 3$

  • $5x - 5y = 1$

• From first: $x = 3 - y$ Substitute into second: $5(3 - y) - 5y = 1 \implies y = 1.4$, $x = 1.6$

Graphical Method Example

• Solve:

  • $x + y = 3$

  • $2x + y = 4$

• Plot both lines; intersection point is the solution: $(x, y) = (1, 2)$

Word Problem Example

• Cost of 6 trousers and 4 shirts is 37,200 Tshs. Cost of 4 trousers and 3 shirts is 25,500 Tshs. Find the cost of each.

• Let $x$ = trousers, $y$ = shirts. $6x + 4y = 37,200$ $4x + 3y = 25,500$ Solution: $x = 4,800$, $y = 2,100$

Inequalities

Definition and Properties

• Inequality: A statement about the relative size of two values.

Symbol	Words	Example
>	greater than	$x + 3 > 2$
<	less than	$7x < 28$
≥	greater than or equal to	$5 \geq x - 1$
≤	less than or equal to	$2y + 1 \leq 7$

Transformations Producing Equivalent Inequalities

• Add/subtract the same number to both sides.

• Multiply/divide both sides by a positive number (direction unchanged).

• Multiply/divide both sides by a negative number (reverse the inequality direction).

Example: Solving an Inequality

• Solve $2x + 1 \leq 6x - 1$

• Steps:

  • $2x + 1 \leq 6x - 1$

  • $-4x + 1 \leq -1$

  • $-4x \leq -2$

  • $x \geq \frac{1}{2}$ (reverse direction when dividing by negative)

Compound Inequalities

• "And" inequalities: variable must satisfy both conditions.

• "Or" inequalities: variable must satisfy at least one condition.

Example: "And" Compound Inequality

• Solve $-2 \leq 3t - 8 \leq 10$

• Steps:

  • $6 \leq 3t \leq 18$

  • $2 \leq t \leq 6$

Example: "Or" Compound Inequality

• Solve $2x + 3 < 5$ or $4x - 7 > 9$

• First: $x < 1$; Second: $x > 4$

Systems of Inequalities

• The solution is the region where all inequalities overlap on the graph.

• Example: $x + y \leq 3$, $2x + y \geq 4$

• Graph both lines and shade the intersection region.

Linear Programming

Definition and Applications

• Linear programming: A mathematical technique for optimizing (maximizing or minimizing) a linear objective function, subject to linear constraints.

• Applications: resource allocation, manufacturing, finance, network design, etc.

Formulating a Linear Programming Problem

1. Define the objective (e.g., maximize profit, minimize cost).

2. Identify decision variables (e.g., number of tables and chairs).

3. List constraints (e.g., time, resources).

4. Write the objective function and constraints as equations/inequalities.

Example: Furniture Production

Department	Decision variable	Resource available
Assembly	Tables: 4, Chairs: 2	60
Finishing	Tables: 2, Chairs: 4	48
Profits (Tshs)	Tables: 6,500, Chairs: 4,500

• Let $x$ = number of tables, $y$ = number of chairs

• Objective function: Maximize $Z = 6,500x + 4,500y$

• Constraints:

  • $4x + 2y \leq 60$

  • $2x + 4y \leq 48$

  • $x \geq 0, y \geq 0$

Solving by Graphical Method

1. Plot constraints on a Cartesian plane.

2. Identify the feasible region (satisfies all constraints).

3. Find coordinates of extreme points (vertices) of the feasible region.

4. Evaluate the objective function at each vertex.

5. The optimal solution is at the vertex with the highest (or lowest) value, depending on the problem.

Functions

Definition and Notation

• A function is a correspondence between two sets, where each input (domain) has exactly one output (range).

• Notation: $f(x)$, $y = f(x)$

• Example: $f(x) = 3x + 1$

Examples of Functions

• $f(x) = 3x^2 + 4x - 1$

• $h(x) = 3x + ax^2$

Determining if an Equation is a Function

• Each $x$ must correspond to only one $y$.

• Equations like $y^2 = x$ are not functions (one $x$ gives two $y$ values).

Domain and Range

• Domain: Set of all possible input values ($x$).

• Range: Set of all possible output values ($y$).

• Example: For $f(x) = 2x + 5$, domain and range are all real numbers.

Graphing Functions

• Find $x$- and $y$-intercepts.

• Check for symmetry and asymptotes.

• Determine maximum or minimum values (for quadratics, use vertex formula).

Common Types of Functions

• Polynomial functions (constant, linear, quadratic, etc.)

• Rational functions

• Step functions

• Periodic functions

• Absolute value functions

• Exponential and logarithmic functions

• Trigonometric functions

Linear Functions

• Form: $f(x) = ax + b$

• Graph is a straight line.

• Domain and range: all real numbers.

Quadratic Functions

• Form: $f(x) = ax^2 + bx + c$, $a \neq 0$

• Graph is a parabola (opens up if $a > 0$, down if $a < 0$).

• Vertex (maximum or minimum): $x = -\frac{b}{2a}$

Application Example: Cost and Revenue

• Cost function: $C(x) = 20x - 5$

• Revenue function: $R(x) = x^2 + 20$

• Intersection points: solve $C(x) = R(x)$

Applications of Linear and Quadratic Functions

• Supply and Demand:

  • Demand function: $D = a - bP$

  • Supply function: $S = -a + bP$

  • Equilibrium: set $D = S$ and solve for $P$

• Break-even analysis: Find where cost equals revenue.