Question

1. \\(\frac{7}{3} - \frac{8}{12x - 8}\\)\
2. \\(\frac{5}{n + 5} + \frac{4n}{2n + 6}\\)\
3. \\(\frac{2x}{5x + 4} + \frac{6x}{2x + 3}\\)

Explanation:

Problem 7

Step1: Factor denominator

$12x-8=4(3x-2)$

Step2: Find common denominator

Common denominator is $4(3x-2)$

Step3: Rewrite first fraction

$\frac{7}{3}=\frac{7\times4(3x-2)}{3\times4(3x-2)}=\frac{28(3x-2)}{12(3x-2)}$

Step4: Rewrite second fraction

$\frac{8}{12x-8}=\frac{8\times3}{4(3x-2)\times3}=\frac{24}{12(3x-2)}$

Step5: Subtract fractions

$\frac{28(3x-2)-24}{12(3x-2)}=\frac{84x-56-24}{12(3x-2)}=\frac{84x-80}{12(3x-2)}$

Step6: Simplify numerator/denominator

$\frac{4(21x-20)}{4\times3(3x-2)}=\frac{21x-20}{3(3x-2)}$

Problem 8

Step1: Factor denominator

$2n+6=2(n+3)$

Step2: Find common denominator

Common denominator is $2(n+3)(n+5)$

Step3: Rewrite first fraction

$\frac{5}{n+5}=\frac{5\times2(n+3)}{2(n+3)(n+5)}=\frac{10(n+3)}{2(n+3)(n+5)}$

Step4: Rewrite second fraction

$\frac{4n}{2n+6}=\frac{4n(n+5)}{2(n+3)(n+5)}$

Step5: Add fractions

$\frac{10(n+3)+4n(n+5)}{2(n+3)(n+5)}$

Step6: Expand numerator

$10n+30+4n^2+20n=4n^2+30n+30$

Step7: Simplify numerator

$\frac{2(2n^2+15n+15)}{2(n+3)(n+5)}=\frac{2n^2+15n+15}{(n+3)(n+5)}$

Problem 9

Step1: Factor denominator

$6x+4=2(3x+2)$

Step2: Find common denominator

Common denominator is $2(3x+2)(2x+3)$

Step3: Rewrite first fraction

$\frac{2x}{6x+4}=\frac{2x(2x+3)}{2(3x+2)(2x+3)}$

Step4: Rewrite second fraction

$\frac{6x}{2x+3}=\frac{6x\times2(3x+2)}{2(3x+2)(2x+3)}=\frac{12x(3x+2)}{2(3x+2)(2x+3)}$

Step5: Add fractions

$\frac{2x(2x+3)+12x(3x+2)}{2(3x+2)(2x+3)}$

Step6: Expand numerator

$4x^2+6x+36x^2+24x=40x^2+30x$

Step7: Simplify numerator

$\frac{10x(4x+3)}{2(3x+2)(2x+3)}=\frac{5x(4x+3)}{(3x+2)(2x+3)}$

Answer:

1. $\frac{21x-20}{3(3x-2)}$
2. $\frac{2n^2+15n+15}{(n+3)(n+5)}$
3. $\frac{5x(4x+3)}{(3x+2)(2x+3)}$