College Algebra: Equations and Inequalities – Study Notes

Equations and Inequalities

Introduction

This chapter introduces the foundational concepts of equations and inequalities, focusing on linear equations, procedures for solving them, and their applications. Mastery of these topics is essential for success in College Algebra and further mathematical studies.

Objectives

• Solve a Linear Equation: Learn systematic methods for finding the value of the variable that satisfies a linear equation.

• Solve Equations That Lead to Linear Equations: Transform and simplify equations to the linear form for easier solution.

• Solve Problems That Can Be Modeled by Linear Equations: Apply algebraic techniques to real-world scenarios.

Procedures That Result in Equivalent Equations

Basic Operations

Equivalent equations have the same solution set. The following operations can be performed to transform equations while preserving equivalence:

• Interchange Sides: The sides of an equation can be swapped without affecting the solution. Example: Replace $3 = x$ by $x = 3$.

• Simplify Sides: Combine like terms and eliminate parentheses. Example: Replace $(x + 2) + 6 = 2x + (x + 1)$ by $x + 8 = 3x + 1$.

• Add/Subtract the Same Expression: Adding or subtracting the same value from both sides keeps the equation equivalent. Example: Replace $3x - 5 = 4$ by $(3x - 5) + 5 = 4 + 5$.

• Multiply/Divide by Nonzero Expression: Multiplying or dividing both sides by the same nonzero value preserves equivalence. Example: Replace $\frac{3x}{x-1} = \frac{6}{x-1}$ by $3x = 6$ (for $x \neq 1$).

• Factoring: If one side is zero and the other can be factored, write as a product of factors. Example: Replace $x^2 - 3x = 0$ by $x(x - 3) = 0$.

Solving Linear Equations

Definition

A linear equation in one variable is an equation equivalent in form to $ax + b = 0$, where $a$ and $b$ are real numbers and $a \neq 0$.

General Solution

• Isolate the variable $x$:

$ax + b = 0$ Subtract $b$ from both sides: $ax = -b$ Divide both sides by $a$: $x = -\frac{b}{a}$

Example: Solving a Linear Equation

• Given: $4x - 5 = 7$

• Add 5 to both sides: $4x - 5 + 5 = 7 + 5$

• Simplify: $4x = 12$

• Divide both sides by 4: $x = 3$

Check: Substitute $x = 3$ into the original equation: $4(3) - 5 = 7$ $12 - 5 = 7$ $7 = 7$

Steps for Solving Equations

1. List Restrictions: Identify any restrictions on the domain of the variable (e.g., denominators cannot be zero).

2. Simplify: Use equivalent equations to simplify.

3. Zero-Product Property: If the equation is a product of factors equal to zero, set each factor to zero and solve.

4. Check Solution(s): Substitute back to verify correctness.

Applications of Linear Equations

Modeling Real-World Problems

• Assign variables to unknowns.

• Translate the problem into an equation.

• Solve the equation.

• Interpret the solution in context.

Example: Investment Problem Suppose $10,000 is invested in stocks and bonds. If the amount in bonds is one-fourth that in stocks, let$$s$ be the amount in stocks. Then, amount in bonds is $10,000 - s$ and $10,000 - s = \frac{1}{4}s$. Solve for $s$:

$10,000 - s = \frac{1}{4}s$ $10,000 = s + \frac{1}{4}s$ $10,000 = \frac{5}{4}s$ $s = 8,000$ Amount in bonds: $10,000 - 8,000 = 2,000$

Summary Table: Procedures for Equivalent Equations

Operation	Example	Purpose
Interchange Sides	$3 = x$ to $x = 3$	Rewriting for convenience
Simplify Sides	$(x + 2) + 6 = 2x + (x + 1)$ to $x + 8 = 3x + 1$	Combine like terms
Add/Subtract	$3x - 5 = 4$ to $3x = 9$	Isolate variable
Multiply/Divide	$\frac{3x}{x-1} = \frac{6}{x-1}$ to $3x = 6$	Clear denominators
Factoring	$x^2 - 3x = 0$ to $x(x - 3) = 0$	Apply Zero-Product Property

Key Terms

• Linear Equation: An equation of the form $ax + b = 0$.

• Equivalent Equations: Equations with the same solution set.

• Zero-Product Property: If $ab = 0$, then $a = 0$ or $b = 0$.

• Domain: The set of permissible values for the variable.

Conclusion

Understanding how to manipulate and solve linear equations is a fundamental skill in algebra. The procedures outlined above provide a systematic approach to finding solutions and applying these concepts to real-world problems.