Problem solving involving rational equation

1. the reciprocal of 5 plus the reciprocal of 7 is the reciprocal of what number?

2.the sum of a number and 6 times its reciprocal is -5. find the number.

3. the reciprocal of the product of two consecutive intergers is 1/72.

4. the reciprocal of the product of two consecutive integers is 1/42

Step-by-step explanation:

1) 1/5 + 1/7 = 1*7/5*7 + 1*5/7*5

=7/35 + 5/35 = 7+5/35 = 12/35

2) Let the number be x

x +6*1/x = -5

x + 6/x = -5

(x² + 6 ) / x = -5

x² + 6 = -5x

x² +5x + 6 = 0

x² + 3x +2x + 2*3 = 0

x(x+3) + 2(x+3)= 0

(x+3)(x+2) = 0

x + 3 = 0  or x +2 = 0

x = -3    or x = -2

the number is (-3)   or (-2)

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step-by-step explanation:

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