6. Simplify the ff. (a) $$ \frac{x}{4}+\frac{3}{7} $$ b) $$ \frac{2 x}{4+x}+\frac{x}{3} $$ (3) $$ \frac{a+1}{b} imes \frac{b^{2}}{a^{2}-1} $$

Question

6. Simplify the ff. (a) $$ \frac{x}{4}+\frac{3}{7} $$ b) $$ \frac{2 x}{4+x}+\frac{x}{3} $$ (3) $$ \frac{a+1}{b} 	imes \frac{b^{2}}{a^{2}-1} $$

UpStudy AI · Accepted Answer

**(a) Simplify $$ \frac{x}{4} + \frac{3}{7} $$**

1. Find the least common denominator (LCD) of 4 and 7, which is $$ 28 $$.
2. Rewrite each fraction with the denominator $$ 28 $$:
   $$
   \frac{x}{4} = \frac{7x}{28}, \quad \frac{3}{7} = \frac{12}{28}
   $$
3. Combine the fractions:
   $$
   \frac{7x}{28} + \frac{12}{28} = \frac{7x+12}{28}
   $$

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**(b) Simplify $$ \frac{2x}{4+x}+\frac{x}{3} $$**

1. Identify the common denominator for the fractions. The denominators are $$ 4+x $$ and $$ 3 $$. Their LCD is $$ 3(4+x) $$.
2. Rewrite each fraction with the common denominator:
   $$
   \frac{2x}{4+x} = \frac{2x \cdot 3}{3(4+x)} = \frac{6x}{3(4+x)}
   $$
   $$
   \frac{x}{3} = \frac{x(4+x)}{3(4+x)}
   $$
3. Combine the numerators:
   $$
   \frac{6x + x(4+x)}{3(4+x)} = \frac{6x + 4x + x^2}{3(4+x)} = \frac{x^2 + 10x}{3(4+x)}
   $$
4. Factor the numerator if desired:
   $$
   \frac{x(x+10)}{3(4+x)}
   $$
   
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**(3) Simplify $$ \frac{a+1}{b} \times \frac{b^{2}}{a^{2}-1} $$**

1. Factor the difference of squares in the denominator:
   $$
   a^2 - 1 = \left(a+1\right)\left(a-1\right)
   $$
2. Substitute the factorization into the expression:
   $$
   \frac{a+1}{b} \times \frac{b^{2}}{(a+1)(a-1)}
   $$
3. Cancel the common factor $$(a+1)$$:
   $$
   \frac{b^{2}}{b(a-1)}
   $$
4. Simplify by canceling one $$ b $$:
   $$
   \frac{b}{a-1}
   $$

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**Final Answers:**

(a) $$ \frac{7x+12}{28} $$  
(b) $$ \frac{x^2+10x}{3(4+x)} $$ or $$ \frac{x(x+10)}{3(4+x)} $$  
(3) $$ \frac{b}{a-1} $$
(a) $$ \frac{7x + 12}{28} $$ (b) $$ \frac{x^2 + 10x}{3(4 + x)} $$ or $$ \frac{x(x + 10)}{3(4 + x)} $$ (3) $$ \frac{b}{a - 1} $$

Simplify the ff.
(a)
b)
(3)

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