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Ch. 1 - Equations and Inequalities
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 1 - Equations and InequalitiesProblem 8
Chapter 2, Problem 8

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

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1
Recall that a contradiction is an equation that has no solution, meaning it is never true for any value of the variable.
Start with the given equation: \(5x = 4x\).
Subtract \$4x$ from both sides to isolate the variable terms on one side: \(5x - 4x = 4x - 4x\).
Simplify both sides: \(x = 0\).
Since the equation simplifies to \(x = 0\), which is a specific solution, the equation is not a contradiction because it has at least one solution.

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Key Concepts

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

Equation and Solution

An equation is a mathematical statement asserting the equality of two expressions. Solving an equation involves finding all values of the variable that make the equation true. Understanding what it means for an equation to have solutions is fundamental to analyzing its nature.
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Contradiction in Equations

A contradiction is an equation that has no solution because it leads to a false statement, such as 0 = 5. Recognizing contradictions helps determine when an equation cannot be satisfied by any value of the variable.
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Simplifying and Comparing Expressions

Simplifying both sides of an equation by combining like terms or performing operations helps reveal the true nature of the equation. For example, simplifying 5x = 4x leads to x = 0, showing whether the equation is always true, false, or conditional.
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Related Practice
Textbook Question

Match each equation or inequality in Column I with the graph of its solution set in Column II. | x | ≠ 7

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Textbook Question

Match each equation or inequality in Column I with the graph of its solution set in Column II. | x | ≤ 7

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Textbook Question

Use Choices A–D to answer each question. A. 3x2 - 17x - 6 = 0 B. (2x + 5)2 = 7 C. x2 + x = 12 D. (3x - 1)(x - 7) = 0 Which equation is set up for direct use of the zero-factor property? Solve it.

Textbook Question

Decide whether each statement is true or false. If false, correct the right side of the equation. i12 = 1

Textbook Question

Match each equation in Column I with the correct first step for solving it in Column II. √(x+5) = 7

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Textbook Question

Match each equation in Column I with the correct first step for solving it in Column II. (x+5)2/3 - (x+5)1/3 - 6 = 0