Question

b. calculate each sum and difference. simplify the answer when possible.

1. \\(\frac{x}{2} + \frac{7x}{6}\\)
2. \\(\frac{x + 2}{4} - \frac{1}{10}\\)
3. \\(\frac{-2x}{5} - \frac{y}{10} + z\\)
4. \\(\frac{x - 3}{15} - \frac{x - 3}{10}\\)
5. \\(\frac{2x}{7} - \frac{y}{2} + \frac{x + 1}{3}\\)
6. \\(\frac{x - 1}{4} + \frac{x + 2}{2} - \frac{x^2}{8}\\)
7. \\(\frac{2x}{5} + \frac{2x^2 - 1}{10} - \frac{4x + 1}{6}\\)
8. \\(\frac{x + 4}{2} - \frac{x + 5}{3} + \frac{x + 6}{21}\\)

Explanation:

Problem 2

Step1: Find common denominator

The least common denominator of 4 and 10 is 20.

Step2: Rewrite fractions

$\frac{x+2}{4} = \frac{5(x+2)}{20}$, $\frac{z}{10} = \frac{2z}{20}$

Step3: Subtract fractions

$\frac{5(x+2) - 2z}{20} = \frac{5x + 10 - 2z}{20}$

Problem 3

Step1: Find common denominator

The least common denominator of 5, 10, and 1 (for $z$) is 10.

Step2: Rewrite fractions

$\frac{-2x}{5} = \frac{-4x}{10}$, $\frac{y}{10} = \frac{y}{10}$, $z = \frac{10z}{10}$

Step3: Combine terms

$\frac{-4x - y + 10z}{10}$

Problem 4

Step1: Find common denominator

The least common denominator of 15 and 10 is 30.

Step2: Rewrite fractions

$\frac{x-3}{15} = \frac{2(x-3)}{30}$, $\frac{x-3}{10} = \frac{3(x-3)}{30}$

Step3: Subtract fractions

$\frac{2(x-3) - 3(x-3)}{30} = \frac{(2x-6)-(3x-9)}{30}$

Step4: Simplify numerator

$\frac{2x - 6 - 3x + 9}{30} = \frac{-x + 3}{30} = \frac{-(x-3)}{30}$

Problem 5

Step1: Find common denominator

The least common denominator of 7, 2, and 3 is 42.

Step2: Rewrite fractions

$\frac{2x}{7} = \frac{12x}{42}$, $\frac{y}{2} = \frac{21y}{42}$, $\frac{x+1}{3} = \frac{14(x+1)}{42}$

Step3: Combine terms

$\frac{12x - 21y + 14(x+1)}{42} = \frac{12x - 21y + 14x + 14}{42}$

Step4: Simplify numerator

$\frac{26x - 21y + 14}{42}$

Problem 6

Step1: Find common denominator

The least common denominator of 4, 2, and 8 is 8.

Step2: Rewrite fractions

$\frac{x-1}{4} = \frac{2(x-1)}{8}$, $\frac{x+2}{2} = \frac{4(x+2)}{8}$, $\frac{x^2}{8} = \frac{x^2}{8}$

Step3: Combine terms

$\frac{2(x-1) + 4(x+2) - x^2}{8} = \frac{2x-2 + 4x+8 - x^2}{8}$

Step4: Simplify numerator

$\frac{-x^2 + 6x + 6}{8}$

Problem 7

Step1: Find common denominator

The least common denominator of 5, 10, and 6 is 30.

Step2: Rewrite fractions

$\frac{2x}{5} = \frac{12x}{30}$, $\frac{2x^2-1}{10} = \frac{3(2x^2-1)}{30}$, $\frac{4x+1}{6} = \frac{5(4x+1)}{30}$

Step3: Combine terms

$\frac{12x + 3(2x^2-1) - 5(4x+1)}{30} = \frac{12x + 6x^2 - 3 - 20x - 5}{30}$

Step4: Simplify numerator

$\frac{6x^2 - 8x - 8}{30} = \frac{3x^2 - 4x - 4}{15}$

Problem 8

Step1: Find common denominator

The least common denominator of 2, 3, and 21 is 42.

Step2: Rewrite fractions

$\frac{x+4}{2} = \frac{21(x+4)}{42}$, $\frac{x+5}{3} = \frac{14(x+5)}{42}$, $\frac{x+6}{21} = \frac{2(x+6)}{42}$

Step3: Combine terms

$\frac{21(x+4) - 14(x+5) + 2(x+6)}{42} = \frac{21x+84 -14x-70 +2x+12}{42}$

Step4: Simplify numerator

$\frac{9x + 26}{42}$

Answer:

1. $\frac{5x}{3}$
2. $\frac{5x + 10 - 2z}{20}$
3. $\frac{-4x - y + 10z}{10}$
4. $\frac{-x + 3}{30}$
5. $\frac{26x - 21y + 14}{42}$
6. $\frac{-x^2 + 6x + 6}{8}$
7. $\frac{3x^2 - 4x - 4}{15}$
8. $\frac{9x + 26}{42}$