Let’s solve each rational expression step by step. We’ll combine like terms, simplify numerators, and reduce if possible.
---
Problem 1:
\[
\frac{7x + 4}{x + 2} + \frac{5x - 7}{x + 2}
\]
Same denominator → add numerators:
\[
= \frac{(7x + 4) + (5x - 7)}{x + 2} = \frac{12x - 3}{x + 2}
\]
Check if numerator factors:
12x - 3 = 3(4x - 1) — no common factor with denominator → leave as is.
✔ Final for Problem 1: \(\frac{12x - 3}{x + 2}\)
---
Problem 2:
\[
\frac{7}{x} - \frac{11}{x}
\]
Same denominator → subtract numerators:
\[
= \frac{7 - 11}{x} = \frac{-4}{x}
\]
✔ Final for Problem 2: \(-\frac{4}{x}\)
---
Problem 3:
\[
\frac{3x - 10}{x - 1} - \frac{5x - 12}{x - 1}
\]
Same denominator → subtract numerators:
\[
= \frac{(3x - 10) - (5x - 12)}{x - 1}
\]
Distribute the minus sign:
\[
= \frac{3x - 10 - 5x + 12}{x - 1} = \frac{-2x + 2}{x - 1}
\]
Factor numerator:
\[
= \frac{-2(x - 1)}{x - 1}
\]
Cancel \(x - 1\) (as long as \(x ≠ 1\)):
\[
= -2
\]
✔ Final for Problem 3: \(-2\)
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Problem 4:
\[
\frac{3x}{5} + \frac{4x + 7}{5}
\]
Same denominator → add numerators:
\[
= \frac{3x + 4x + 7}{5} = \frac{7x + 7}{5}
\]
Factor numerator:
\[
= \frac{7(x + 1)}{5}
\]
No further simplification.
✔ Final for Problem 4: \(\frac{7x + 7}{5}\) or \(\frac{7(x + 1)}{5}\)
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Final Answer:
Problem 1: \(\frac{12x - 3}{x + 2}\)
Problem 2: \(-\frac{4}{x}\)
Problem 3: \(-2\)
Problem 4: \(\frac{7x + 7}{5}\)
Parent Tip: Review the logic above to help your child master the concept of algebra 2 adding and subtracting rational expressions worksheet answers.
