Math 017 Materials With Exercises

■ Solve the following equation 2x 1 2x 5.2x 1 2x 5Subtract 2 x from both sides. 1 5As we know, subtracting 2 x from both sides produces an equation with exactly the same solution setas the original one. But 1 5 is a false statement. No matter what the value of x is, it is never true.In other words, there is no value of x that would satisfy it. We conclude:The equation 2x 1 2x 5 does not have a solution. We write „no solution‟ as the answer. ■■ Solve the following equation 2(x 3) 2x 6 .2(x 3) 2x 6Remove parentheses.2x 6 2x 6Subtract 2 x from both sides. 6 6The statement 6 6is true. It is true regardless of the value of x . It is true if x 1,x 2,orx 1000. It is true for any value of x . All real numbers make it true. Therefore, the solution of2(x 3) 2x 6 is all real numbers. We write „all real numbers‟ as the answer. ■In general, if in the process of solving an equation the variable is cancelled (disappears from theequation), then the equation either has no solution or its solution is all real numbers.If an equation reduces to a false numerical statement, the original equation does not havea solution.If an equation reduces to a true numerical statement, the original equation has a solutionconsisting of all real numbers.Example 8.5 Solve the following equation 4x 4( x 1)Solution:4x 4( x 1)Remove parentheses using the Distributive Law.4x 4x 4 Subtract 4 x from both sides.0 4Since the statement 0 4is false, the equation 4x 4( x 1)has no solutions.Common mistakes and misconceptionsMistake 8.1Algebraic expressions cannot be ‘solved’. To solve means to find all values of variables forwhich the left-hand side of the equation is equal to the right-hand side. So, for example, itwould be meaningless to use the phrase ‘Solve 3 x + 2’.83