Problem 8

Solve the equation 5⁢xx+12-8-3⁢x3⁢x-1=1. the fraction with numerator 5 x and denominator x plus 12 minus the fraction with numerator 8 minus 3 x and denominator 3 x minus 1 equals 1 period

Solution

a) Multiply both sides by the common denominator (x+12)⁢(3x-1) open paren x plus 12 close paren times open paren number 3 x minus 1 close paren:

5⁢x⁢(3⁢x-1)-(8-3⁢x)⁢(x+12)=(x+12)⁢(3⁢x-1) 5 x times open paren 3 x minus 1 close paren minus open paren 8 minus 3 x close paren times open paren x plus 12 close paren equals open paren x plus 12 close paren times open paren 3 x minus 1 close paren

b) Expand each term. Compute:

5⁢x⁢(3⁢x-1)=15⁢x2-5⁢x 5 x times open paren 3 x minus 1 close paren equals 15 x squared minus 5 x

and

(8-3⁢x)⁢(x+12)=8⁢x+96-3⁢x2-36⁢x=-3⁢x2-28⁢x+96. open paren 8 minus 3 x close paren times open paren x plus 12 close paren equals 8 x plus 96 minus 3 x squared minus 36 x equals negative 3 x squared minus 28 x plus 96 period

Since the second term is subtracted, it becomes

+3⁢x2+28⁢x-96. positive 3 x squared plus 28 x minus 96 period

Thus, the left-hand side becomes

15⁢x2-5⁢x+3⁢x2+28⁢x-96=18⁢x2+23⁢x-96. 15 x squared minus 5 x plus 3 x squared plus 28 x minus 96 equals 18 x squared plus 23 x minus 96 period

The right-hand side expands to

(x+12)⁢(3⁢x-1)=3⁢x2+35⁢x-12. open paren x plus 12 close paren times open paren 3 x minus 1 close paren equals 3 x squared plus 35 x minus 12 period

c) Form the equation: 18⁢x2+23⁢x-96=3⁢x2+35⁢x-12. 18 x squared plus 23 x minus 96 equals 3 x squared plus 35 x minus 12 period

Subtract the right-hand side from the left:

15⁢x2-12⁢x-84=0. 15 x squared minus 12 x minus 84 equals 0 period

d) Divide by 3: 5⁢x2-4⁢x-28=0. 5 x squared minus 4 x minus 28 equals 0 period Apply the quadratic formula:

x=4±(-4)2-4⋅5⋅(-28)2⋅5=4±16+56010=4±57610 x equals the fraction with numerator 4 plus or minus the square root of open paren negative 4 close paren squared minus 4 times 5 times negative 28 and denominator 2 times 5 equals the fraction with numerator 4 plus or minus the square root of 16 plus 560 and denominator 10 equals the fraction with numerator 4 plus or minus the square root of 576 and denominator 10

Since 576=24 the square root of 576 equals 24, we have x=4±2410 x equals the fraction with numerator 4 plus or minus 24 and denominator 10

Thus,

x=2810=145 x equals 28 over 10 equals fourteen fifths or x=-2010=-2. x equals negative 20 over 10 equals negative 2 period

e) Check that x≠-12 x is not equal to negative 12 and x≠13 x is not equal to one third from the original denominators; both solutions are acceptable.

Answer: x=145 x equals fourteen fifths or x=-2. x equals negative 2 period