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Second Edition: 2012-2013
7.4.SOLVING RATIONAL EQUATIONS45123.The least common denominator (LCD) isx2, so first clear fractions bymultiplying both sides of the equation by the LCD.2x= 3 +44xOriginal equation.x[2x] =3 +44xxMultiply both sides byx.x[2x] =x +x44xDistributex.2x2= 3x+ 44Cancel and simplify.The resulting equation is nonlinear (xis raised to a power larger than 1). Makeone side zero2x2−3x= 44Subtract 3xfrom both sides.2x2−3x−44 = 0Subtract 44 from both sides.Compare 2x2−3x−44 withax2+bx+cand note thatac= (2)(−44) andb=−3. The integer pair−11 and 8 have productac=−88 and sumb=−3.Replace the middle term with a combination of like terms using this pair, thenfactor by grouping.2x2−11x+ 8x−44 = 0−3x=−11x+ 8xx(2x−11) + 4(2x−11) = 0Factor by grouping.(x+ 4)(2x−11) = 0Factor out 2x−11.Use the zero product property to complete the solution. Either the first factoris zero or the second factor is zero.x+ 4 = 0or2x−11 = 0x=−4x=112Hence, the solutions arex=−4 andx= 11/2.Graphical solution:Make one side of the equation zero.2x−3−44x= 0Load the left-hand side of the equation intoY1as2*X-3-44/X(see imageon the left), then select6:ZStandardfrom the ZOOM menu to produce theimage at the right.Second Edition: 2012-2013
452CHAPTER 7.RATIONAL FUNCTIONSNext, the solutions of2x−3−44x= 0are found by noting where the graph ofy= 2x−3−44/xcross thex-axis. Select2:zerofrom the CALC menu. Use the arrow keys to move the cursor to theleft of the firstx-intercept, then press ENTER to set the “Left bound.” Next,move the cursor to the right of the firstx-intercept, then press ENTER to setthe “Right bound.” Finally, leave the cursor where it is and press ENTER toset your “Guess.” The calculator responds with the result shown in the figureon the left.Repeat the zero-finding procedure to capture the coordinates ofthe secondx-intercept (see the image on the right).Reporting the solution on your homework.Second Edition: 2012-2013
7.4.SOLVING RATIONAL EQUATIONS453−1010−1010xy−45.5y= 2x−3−44/xNote that the calculator solutions,−4 and 5.5, match the algebraic solutions.25.In the solution, we address each step of theRequirements for Word ProblemSolutions.1.Set up a variable dictionary.Letxrepresent the unknown number.2.Set up an equation.If the unknown number isx, then its reciprocal is1/x. Thus, the “sum of a number and its reciprocal is 5/2” becomes:x+1x=523.Solve the equation.Clear the fractions by multiplying both sides by 2x,the least common denominator.x+1x=52Model equation.2xx+1x=522xMultiply both sides by 2x.2x[x] + 2x1x=522xDistribute 2x.2x2+ 2 = 5xCancel and simplify.The equation is nonlinear. Make one side zero.2x2−5x+ 2 = 0Make one side zero.