130 Chapter 2 Linear Equations and Inequalities in One Variable ������������������������ An equation that is not true for any value of the variable is called a contradiction. A contradiction has no solution. The solution set is the empty set, or null set, ∅. Contradictions do not initially look any different from conditional equations. The difference becomes apparent in the equation-solving process. As we solve the equation, the variable terms are eliminated from both sides of the equation and a false statement (e.g., 5 = 7) remains. If we solve an equation and obtain a false statement, then the equation is a contradiction and has no solution. The solution set is the empty set, ∅. �� ������������������������ ������������������ Solve each equation. 3a. x + 3 = x + 2 3b. 4x + 5 - x = 3(x + 2) Solutions 3a. x + 3 = x +2 x + 3 - x = x + 2 - x Subtract x from each side. 3 =2 Simplify. The resulting equation, 3 = 2, is a contradiction, or false statement. So, the solution set is the empty set, ∅. 3b. 4x + 5 - x = 3(x + 2) 3x + 5 = 3x + 6 3x + 5 - 3x = 3x + 6 - 3x 5 = 6 Apply the distributive property on the right side and combine like terms on the left side. Subtract 3x from each side. Simplify. The resulting equation, 5 = 6, is a contradiction, so the solution set is the empty set, ∅. Student Check 3 Solve each equation. a. 2x - 4 = 2x + 1 b. 5(2x - 4) = 3(3x + 1) + x ������������ Once an equation is simplified on each side, we know we have a contradiction when the variable term on each side of the equation is the same but the constant terms are different. Linear Equations with Infinitely Many Solutions We will now turn our attention to solving linear equations that have infinitely many solutions. Consider the equation x + 1 = x + 1. The left and right sides of this equation will always be equal since the right side of the equation is the exact same as the left side of the equation as shown in the following table. x x + 1 = x + 1 -2 -2 + 1 = -2 + 1 -1 = -1 True -1 -1 + 1 = -1 + 1 0 = 0 True 0 0 + 1 = 0 + 1 1 = 1 True 1 1 + 1 = 1 + 1 2 = 2 True 2 2 + 1 = 2 + 1 3 = 3 True Objective 4 ▶ Solve linear equations with infinitely many solutions.
hendricks_beginning_algebra_1e_ch1_3
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