52 Chapter 1 Linear Equations and Inequalities in One Variable 4. Conditional Equations, Contradictions, x 1 x 2 1 2 1contradiction2 This indicates that the equation has no solution. An equation that has no solution is called a contradiction.The solution set for a contradiction is the empty set.The empty set is the set with no elements and is denoted by or . III. Identities An equation that is true for all real numbers is called an identity. For example, consider the equation 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 solve the equation, we get the identity In such a case, the solution set is the set of real numbers. x 4 x 4 4 4 1identity2 The solution set is the set of real numbers. In set-builder notation, we have, Identifying Conditional Equations, Contradictions, and Identities Example 10 Identify each equation as a conditional equation, a contradiction, or an identity. Then give the solution set. a. 33x 1x 12 4 2 b. 513 c2 2 2c 3c 17 c. 4x 3 17 5x 0 x is a real number6. x x 4 x x 4 4 4. x 4 x 4. 5 6 x x 1 x x 2 and Identities The solution to a linear equation is the value of x that makes the equation a true statement.While linear equations have one unique solution, some equations have no solution, and others have infinitely many solutions. 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 is a conditional equation because it is true on the condition that For other values of x, the statement is false.The solution set is {2}. II. Contradictions Some equations have no solution, such as x 1 x 2. 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 1 2. x 4 6 x 2. x 4 6
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