302 CHAPTER 6 Algebra, Graphs, and Functions Timely Tip Remember that the goal in solving an equation is to get the variable alone on one side of the equal sign by using the general procedure for solving equations. Example 11 Solving an Equation Containing Decimals Solve the equation − = + x x 4 0.48 0.8 4 and check your solution. Solution This equation may be solved with the decimals, or you may multiply each term by 100 and eliminate the decimals. We will solve the equation with the decimals. x x x x x x x x x x x x x 4 0.48 0.8 4 4 0.48 0.48 0.8 4 0.48 4 0.8 4.48 4 0.8 0.8 0.8 4.48 3.2 4.48 3.2 3.2 4.48 3.2 1.4 − = + − + = + + = + − = − + = = = Addition property Subtraction Property Division property CHECK: − = + − = + − = + = x x 4 0.48 0.8 4 4(1.4) 0.48 0.8(1.4) 4 5.6 0.48 1.12 4 5.12 5.12 ? ? Substitute 1.4 for each x in the equation. True 7 Now try Exercise 67 So far, every equation has had exactly one solution. Some equations, however, have no solution, and others have more than one solution. Example 12 illustrates an equation that has no solution, and Example 13 illustrates an equation that has an infinite number of solutions. Example 12 An Equation with No Solution Solve − + + = − + x x x x 3( 5) 12 6 2( 2). Solution x x x x x x x x x x x x x x 3( 5) 12 6 2( 2) 3 15 12 6 2 4 4 3 4 4 4 4 3 4 4 4 3 4 − + + = − + − + + = − − − = − − − = − − − = − Distributive property Combine like terms. Subtraction property False During the process of solving an equation, if you obtain a false statement like 3 4, − = − or = 1 0, the equation has no solution. 7 Now try Exercise 71 An equation that has no solution is called a contradiction . The equation in Example 12, − + + = − + x x x x 3( 5) 12 6 2( 2), is a contradiction and thus has no solution. Example 13 An Equation with Infinitely Many Solutions Solve + − − = − + x x x 2( 4) 3( 5) 23. Solution + − − = − + + − + = − + − + = − + x x x x x x x x 2( 4) 3( 5) 23 2 8 3 15 23 23 23 Distributive property Combine like terms.
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