answer:Multiplying both sides of the equation by (x-2)(x-4)(x-3), we get [(x-4)(x-3) + (x-2)(x-3) = 3(x-2)(x-4).] Expanding, we have [(x^2 - 7x + 12) + (x^2 - 5x + 6) = 3(x^2 - 6x + 8).] Combining like terms, we get [2x^2 - 12x + 18 = 3x^2 - 18x + 24.] Moving all the terms to one side, we have [x^2 - 6x + 6 = 0.] This quadratic does not factor nicely, so we can use the quadratic formula to find the solutions: [x = frac{6 pm sqrt{(-6)^2 - 4(1)(6)}}{2(1)} = frac{6 pm sqrt{12}}{2} = frac{6 pm 2sqrt{3}}{2}.] Simplifying, we have [x = 3 pm sqrt{3}.] Since we want the smallest value of x, the answer is boxed{3 - sqrt{3}}.The answer is: 3-sqrt{3}

answer:Tommy has 3 toy cars. Jessie has 3 cars too. Jessie's older brother has x more cars than Tommy and Jessie, which means he has 3 + 3 + x = 6 + x cars. The total number of cars they have altogether is the sum of the number of cars Tommy, Jessie, and Jessie's older brother have: 3 + 3 + 6 + x = 12 + x. We are given that the total number of cars they have altogether is 17, so we can write: 12 + x = 17. Solving for x, we get: x = 5. The value of x is 5. 5 The answer is: 5

answer:The cost of purchasing 38 pencils at 2.50 each is 38 * 2.50 = <<38*2.50=95>>95. The cost of purchasing 56 pens at 3.50 each is 56 * 3.50 = <<56*3.50=196>>196. Therefore, the total cost of purchasing the pencils and pens is 95 + 196 = <<95+196=291>>291. 291 The answer is: 291

answer:By Vieta's formulas, the sum of the roots of the polynomial x^2 - 5x + t is 5. Since the roots are positive integers, the possible pairs of roots are (1,4) and (2,3). For the pair (1,4), the value of t is 1 cdot 4 = 4. For the pair (2,3), the value of t is 2 cdot 3 = 6. Thus, the distinct possible values of t are 4 and 6, and their average is (4+6)/2 = boxed{5}. The answer is: 5