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## 8.6 Solve Rational Equations

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**8.6 Solve Rational Equations**VOCABULARY Cross multiplying**VOCABULARYCross multiplying**A method of solving a simple rational equation for which each side of the equation is a single rational expression. Equal products are formed by multiplying the numerator of each expression by the denominator of the other.**20**5 = - - 2 x 5 x 2 Example 1Solve a rational equation by cross multiplying**20**5 = - - 2 x 5 x 2 Example 1Solve a rational equation by cross multiplying Step 1: Cross Multiply**20**5 = - - 2 x 5 x 2 Example 1Solve a rational equation by cross multiplying Step 1: Cross Multiply 20(x-2) = 5(2x-5)**20**5 = - - 2 x 5 x 2 Example 1Solve a rational equation by cross multiplying Step 1: Cross Multiply 20(x-2) = 5(2x-5) Step 2: Distribute (multiply)**20**5 = - - 2 x 5 x 2 Example 1Solve a rational equation by cross multiplying Step 1: Cross Multiply 20(x-2) = 5(2x-5) Step 2: Distribute (multiply) 20x-40 = 10x-25**Step 3: Move all x’s to the left of the equal sign.**20x-40 = 10x-25 -10x -10x**Step 3: Move all x’s to the left of the equal sign.**20x-40 = 10x-25 -10x -10x 10x - 40 = -25**10x - 40 = -25**Step 4: Get x all by itself**10x - 40 = -25**10x - 40 = -25 +40 +40 Step 4: Get x all by itself**10x - 40 = -25**10x - 40 = -25 +40 +40 Step 4: Get x all by itself 10x = 15**10x - 40 = -25**10x - 40 = -25 +40 +40 Step 4: Get x all by itself 10x = 15 10x = 15 10 10**10x - 40 = -25**10x - 40 = -25 +40 +40 Step 4: Get x all by itself 10x = 15 10x = 15 10 10 x = 1.5**-**- 6 12 = + - x 2 x 1 Checkpoint: 1solve the equation by cross multiplying**-**- 6 12 = + - x 2 x 1 Checkpoint: 1solve the equation by cross multiplying -6(x-1) = -12(x+2)**-**- 6 12 = + - x 2 x 1 Checkpoint: 1solve the equation by cross multiplying -6(x-1) = -12(x+2) -6x+6 = -12x-24**-**- 6 12 = + - x 2 x 1 Checkpoint: 1solve the equation by cross multiplying 6x+6 = -24 -6(x-1) = -12(x+2) -6x+6 = -12x-24**-**- 6 12 = + - x 2 x 1 Checkpoint: 1solve the equation by cross multiplying 6x+6 = -24 6x = -30 -6(x-1) = -12(x+2) -6x+6 = -12x-24**-**- 6 12 = + - x 2 x 1 Checkpoint: 1solve the equation by cross multiplying 6x+6 = -24 6x = -30 -6(x-1) = -12(x+2) x = -5 -6x+6 = -12x-24**Assignment: Pg. 592**#4-12 even**-**8 11 14 + = x 3 x Example 2Solve a rational equation with one solution**-**8 11 14 + = x 3 x Example 2Solve a rational equation with one solution Step 1: Find the LCD for all denominators**-**8 11 14 + = x 3 x Example 2Solve a rational equation with one solution Step 1: Find the LCD for all denominators LCD = 3x**-**8 11 14 + = x 3 x Example 2Solve a rational equation with one solution Step 1: Find the LCD for all denominators Step 2:Multiply every thing by LCD LCD = 3x**-**8 11 14 + = x 3 x Example 2Solve a rational equation with one solution Step 1: Find the LCD for all denominators Step 2:Multiply every thing by LCD 3x(8) + 3x(11) = 3x(-14) x 3 x LCD = 3x**3x(8) + 3x(11) = 3x(-14)**x 3 x Step 3: Simplify**3x(8) + 3x(11) = 3x(-14)**x 3 x Step 3: Simplify 24 + 11x = -42**3x(8) + 3x(11) = 3x(-14)**x 3 x Step 3: Simplify 24 + 11x = -42 Step 4: Solve for x**3x(8) + 3x(11) = 3x(-14)**x 3 x Step 3: Simplify 24 + 11x = -42 Step 4: Solve for x 24 + 11x = -42**3x(8) + 3x(11) = 3x(-14)**x 3 x Step 3: Simplify 24 + 11x = -42 Step 4: Solve for x 24 + 11x = -42 -24 -24**3x(8) + 3x(11) = 3x(-14)**x 3 x Step 3: Simplify 24 + 11x = -42 Step 4: Solve for x 24 + 11x = -42 -24 -24 11x = -66**3x(8) + 3x(11) = 3x(-14)**x 3 x Step 3: Simplify 24 + 11x = -42 Step 4: Solve for x 24 + 11x = -42 -24 -24 11x = -66 11 11**3x(8) + 3x(11) = 3x(-14)**x 3 x Step 3: Simplify 24 + 11x = -42 Step 4: Solve for x 24 + 11x = -42 -24 -24 11x = -66 11 11 X = -6**+**3 x 5 8 = + 2 + + - x 3 ( x 3 )( x 4 ) Example 3Solve a rational equation with two solutions**+**3 x 5 8 = + 2 + + - x 3 ( x 3 )( x 4 ) Example 3Solve a rational equation with two solutions Step 1: LCD**+**3 x 5 8 = + 2 + + - x 3 ( x 3 )( x 4 ) Example 3Solve a rational equation with two solutions Step 1: LCD (x+3)(x-4)**Step 2: Multiply by LCD**(x+3)(x-4)(3x+5)=(x+3)(x-4)(2 + 8 ) (x+3) (x+3)(x-4)**Step 3: Simplify**(x-4)(3x-5) = 2(x+3)(x-4) + 8**Step 3: Simplify**(x-4)(3x-5) = 2(x+3)(x-4) + 8 3x² -17x +20 = 2(x²- x -12) + 8**Step 3: Simplify**(x-4)(3x-5) = 2(x+3)(x-4) + 8 3x² -17x +20 = 2(x²- x -12) + 8 3x² -17x +20 = 2x² -2x -24 + 8**Step 3: Simplify**(x-4)(3x-5) = 2(x+3)(x-4) + 8 3x² -17x +20 = 2(x²- x -12) + 8 3x² -17x +20 = 2x² -2x -24 + 8 3x² -17x +20 = 2x² -2x – 16**Step 4: Solve for x**3x² -17x +20 = 2x² -2x – 16**Step 4: Solve for x**3x² -17x +20 = 2x² -2x – 16 -2x² +2x +16 -2x² +2x +16**Step 4: Solve for x**3x² -17x +20 = 2x² -2x – 16 -2x² +2x +16 -2x² +2x +16 x² -15x +36 = 0**Step 4: Solve for x**3x² -17x +20 = 2x² -2x – 16 -2x² +2x +16 -2x² +2x +16 x² -15x +36 = 0 (x – 3)(x – 12) = 0**Step 4: Solve for x**3x² -17x +20 = 2x² -2x – 16 -2x² +2x +16 -2x² +2x +16 x² -15x +36 = 0 (x – 3)(x – 12) = 0 (x – 3) = 0 or (x – 12) = 0**Step 4: Solve for x**3x² -17x +20 = 2x² -2x – 16 -2x² +2x +16 -2x² +2x +16 x² -15x +36 = 0 (x – 3)(x – 12) = 0 (x – 3) = 0 or (x – 12) = 0 +3 +3 +12 +12**Step 4: Solve for x**3x² -17x +20 = 2x² -2x – 16 -2x² +2x +16 -2x² +2x +16 x² -15x +36 = 0 (x – 3)(x – 12) = 0 (x – 3) = 0 or (x – 12) = 0 +3 +3 +12 +12 x = 3 or x = 12