108 CHAPTER 1 Equations and Inequalities CHECK 2 x + 4 3 + 1 2 x = 1 4 x - 7 3 Original equation 21-42 + 4 3 + 1 2 1-42≟ 1 4 1-42 - 7 3 Let x = -4. -4 3 + 1-22≟-1 - 7 3 Simplify on each side. - 10 3 = - 10 3 ✓ True The solution set is 5-46. S Now Try Exercise 21. Identities, Conditional Equations, and Contradictions An equation satisfied by every number that is a meaningful replacement for the variable is an identity. 31x + 12 = 3x + 3 Identity An equation that is satisfied by some numbers but not others is a conditional equation. 2 x = 4 Conditional equation The equations in Examples 1 and 2 are conditional equations. An equation that has no solution is a contradiction. x = x + 1 Contradiction EXAMPLE 3 Identifying Types of Equations Determine whether each equation is an identity, a conditional equation, or a contradiction. Give the solution set. (a) -21x + 42 + 3x = x - 8 (b) 5x - 4 = 11 (c) 313x - 12 = 9x + 7 SOLUTION (a) -21x + 42 + 3x = x - 8 -2x - 8 + 3x = x - 8 Distributive property x - 8 = x - 8 Combine like terms. 0 = 0 Subtract x. Add 8. When a true statement such as 0 = 0 results, the equation is an identity, and the solution set is {all real numbers}. (b) 5x - 4 = 11 5x = 15 Add 4 to each side. x = 3 Divide each side by 5. This is a conditional equation that is true when x = 3. Its solution set is 536. (c) 313x - 12 = 9x + 7 9x - 3 = 9x + 7 Distributive property -3 = 7 Subtract 9x. When a false statement such as -3 = 7 results, the equation is a contradiction, and the solution set is the empty set, symbolized ∅. S Now Try Exercises 31, 33, and 35.
RkJQdWJsaXNoZXIy NjM5ODQ=