Table of contents

0. Review of Algebra4h 18m

Algebraic Expressions
36m

Exponents
32m

Polynomials Intro
19m

Multiplying Polynomials
36m

Factoring Polynomials
1h 2m

Radical Expressions
15m

Simplifying Radical Expressions
35m

Rationalize Denominator
15m

Rational Exponents
4m

1. Equations & Inequalities3h 18m

Linear Equations
31m

Rational Equations
21m

The Imaginary Unit
6m

Powers of i
11m

Complex Numbers
36m

Intro to Quadratic Equations
18m

The Square Root Property
11m

Completing the Square
12m

The Quadratic Formula
18m

Choosing a Method to Solve Quadratics
9m

Linear Inequalities
20m

2. Graphs of Equations1h 43m

Graphs and Coordinates
7m

Two-Variable Equations
23m

Lines
1h 12m

3. Functions2h 17m

Intro to Functions & Their Graphs
35m

Common Functions
5m

Transformations
45m

Function Operations
21m

Function Composition
29m

4. Polynomial Functions1h 44m

Quadratic Functions
39m

Understanding Polynomial Functions
34m

Graphing Polynomial Functions
30m

Dividing Polynomials

Zeros of Polynomial Functions

5. Rational Functions1h 23m

Introduction to Rational Functions
9m

Asymptotes
35m

Graphing Rational Functions
38m

6. Exponential & Logarithmic Functions2h 28m

Introduction to Exponential Functions
9m

Graphing Exponential Functions
25m

The Number e
8m

Introduction to Logarithms
22m

Graphing Logarithmic Functions
18m

Properties of Logarithms
27m

Solving Exponential and Logarithmic Equations
35m

7. Systems of Equations & Matrices4h 5m

Two Variable Systems of Linear Equations
1h 7m

Introduction to Matrices
1h 11m

Determinants and Cramer's Rule
1h 3m

Graphing Systems of Inequalities
42m

8. Conic Sections2h 23m

Introduction to Conic Sections
5m

Circles
15m

Ellipses: Standard Form
33m

Parabolas
36m

Hyperbolas at the Origin
40m

Hyperbolas NOT at the Origin
12m

9. Sequences, Series, & Induction1h 22m

Sequences
40m

Arithmetic Sequences
25m

Geometric Sequences
15m

10. Combinatorics & Probability1h 45m

Factorials
11m

Combinatorics
46m

Probability
47m

1. Equations & Inequalities

Linear Equations

College Algebra

1. Equations & Inequalities

Linear Equations

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1:28 minutes

Problem 8

Textbook Question

Decide whether each statement is true or false. The equation 5x=4x is an example of a contradiction.

Verified step by step guidance

Understand the definition of a contradiction: A contradiction is an equation that has no solution because it is always false.

Consider the given equation: \(5x = 4x\).

Subtract \(4x\) from both sides of the equation to simplify: \(5x - 4x = 0\).

Simplify the left side: \(x = 0\).

Determine if the equation has a solution: Since \(x = 0\) is a valid solution, the equation is not a contradiction.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Equations and Solutions

An equation is a mathematical statement that asserts the equality of two expressions. To solve an equation, we find the value(s) of the variable that make the equation true. In the case of 5x = 4x, we can simplify it to x = 0, indicating that there is a solution rather than a contradiction.

Contradiction in Mathematics

A contradiction occurs when a statement is proven to be false under all circumstances. In algebra, an example of a contradiction would be an equation that simplifies to a false statement, such as 0 = 5. The equation 5x = 4x does not meet this criterion, as it has a valid solution.

Simplifying Equations

Simplifying an equation involves manipulating it to make it easier to solve or understand. This can include combining like terms, isolating variables, or performing operations on both sides. For 5x = 4x, subtracting 4x from both sides leads to x = 0, demonstrating that the equation is not a contradiction.