136 Chapter 2 Linear Equations and Inequalities I. Conditional Equations An equation that is true for some values of the variable but false for other values is called a conditional equation. The equation x 4 6, for example, is true on the condition that For other values of x, the statement is false. (Conditional equation) The solution is 2. x 4 6 x 4 4 6 4 II. Contradictions Some equations have no solution, such as There is no value of x, that when increased by 1 will equal the same value increased by 2. If we tried to solve the equation by subtracting x from both sides, we get the contradiction This indicates that the equation has no solution. An equation that has no solution is called a contradiction. x 1 x 2 x x 1 x x 2 (Contradiction) No solution. 1 2 III. Identities An equation that has all real numbers as its solution set is called an identity. For example, consider the equation, x 4 x 4. Because the left- and right-hand sides are identical, any real number substituted for x will result in equal quantities on both sides. If we subtract x from both sides of the equation, we get the identity In such a case, the solution is the set of all real numbers. x 4 x 4 4 4. x x 4 x x 4 (Identity) The solution is all real numbers. 4 4 Identifying Conditional Equations, Contradictions, and Identities Example 8 Solve the equation. Identify each equation as a conditional equation, a contradiction, or an identity. a. 4k 5 212k 32 1 b. 21b 42 2b 7 c. 3x 7 2x 5 Solution: a. Clear parentheses. Combine like terms. Subtract 4k from both sides. 4k 5 212k 32 1 4k 5 4k 6 1 4k 5 4k 5 4k 4k 5 4k 4k 5 5 5 1Identity2 This is an identity. The solution is all real numbers. 1 2. x 1 x 2. x 2 x 2. x 4 6 Concept Connections 8. Write a conditional equation. Answers may vary. 9. Write an equation that is a contradiction. Answers may vary. 10. Write an equation that is an identity. Answers may vary. Skill Practice Solve the equation. Identify the equation as a conditional equation, a contradiction, or an identity. 11. 412t 12 1 8t 3 12. 3x 5 4x 1 x 13. 61v 22 2v 4 Classroom Examples: p. 139, Exercises 60, 62, and 64 Answers 8. For example: 9. For example: 10. For example: 11. All real numbers; the equation is an identity. x 5 x 3 x 8 x 9 x 9 12. No solution; the equation is a contradiction. 13. 2; the equation is a conditional equation.
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