Second Edition 2012 2013 74 SOLVING RATIONAL EQUATIONS 451 23 The least common

# Second edition 2012 2013 74 solving rational

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Second Edition: 2012-2013
7.4. SOLVING RATIONAL EQUATIONS 451 23. The least common denominator (LCD) is x 2 , so first clear fractions by multiplying both sides of the equation by the LCD. 2 x = 3 + 44 x Original equation. x [2 x ] = 3 + 44 x x Multiply both sides by x . x [2 x ] = x [3] + x 44 x Distribute x . 2 x 2 = 3 x + 44 Cancel and simplify. The resulting equation is nonlinear ( x is raised to a power larger than 1). Make one side zero 2 x 2 3 x = 44 Subtract 3 x from both sides. 2 x 2 3 x 44 = 0 Subtract 44 from both sides. Compare 2 x 2 3 x 44 with ax 2 + bx + c and note that ac = (2)( 44) and b = 3. The integer pair 11 and 8 have product ac = 88 and sum b = 3. Replace the middle term with a combination of like terms using this pair, then factor by grouping. 2 x 2 11 x + 8 x 44 = 0 3 x = 11 x + 8 x x (2 x 11) + 4(2 x 11) = 0 Factor by grouping. ( x + 4)(2 x 11) = 0 Factor out 2 x 11. Use the zero product property to complete the solution. Either the first factor is zero or the second factor is zero. x + 4 = 0 or 2 x 11 = 0 x = 4 x = 11 2 Hence, the solutions are x = 4 and x = 11 / 2. Graphical solution: Make one side of the equation zero. 2 x 3 44 x = 0 Load the left-hand side of the equation into Y1 as 2*X-3-44/X (see image on the left), then select 6:ZStandard from the ZOOM menu to produce the image at the right. Second Edition: 2012-2013
452 CHAPTER 7. RATIONAL FUNCTIONS Next, the solutions of 2 x 3 44 x = 0 are found by noting where the graph of y = 2 x 3 44 /x cross the x -axis. Select 2:zero from the CALC menu. Use the arrow keys to move the cursor to the left of the first x -intercept, then press ENTER to set the “Left bound.” Next, move the cursor to the right of the first x -intercept, then press ENTER to set the “Right bound.” Finally, leave the cursor where it is and press ENTER to set your “Guess.” The calculator responds with the result shown in the figure on the left. Repeat the zero-finding procedure to capture the coordinates of the second x -intercept (see the image on the right). Reporting the solution on your homework. Second Edition: 2012-2013
7.4. SOLVING RATIONAL EQUATIONS 453 10 10 10 10 x y 4 5 . 5 y = 2 x 3 44 /x Note that the calculator solutions, 4 and 5.5, match the algebraic solutions. 25. In the solution, we address each step of the Requirements for Word Problem Solutions . 1. Set up a variable dictionary. Let x represent the unknown number. 2. Set up an equation. If the unknown number is x , then its reciprocal is 1 /x . Thus, the “sum of a number and its reciprocal is 5/2” becomes: x + 1 x = 5 2 3. Solve the equation. Clear the fractions by multiplying both sides by 2 x , the least common denominator. x + 1 x = 5 2 Model equation. 2 x x + 1 x = 5 2 2 x Multiply both sides by 2 x . 2 x [ x ] + 2 x 1 x = 5 2 2 x Distribute 2 x . 2 x 2 + 2 = 5 x Cancel and simplify. The equation is nonlinear. Make one side zero. 2 x 2 5 x + 2 = 0 Make one side zero.