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Subtracting −2y−2y2+2y−8−y−12−y\frac{-2y-2}{y^2+2y-8} - \frac{y-1}{2-y}

Subtract two rational expressions involving opposite binomial denominators and simplify: −2y−2y2+2y−8−y−12−y\frac{-2y - 2}{y^2 + 2y - 8} - \frac{y - 1}{2 - y}

Step 1 — Factor the denominator and resolve the opposites. Factor the first denominator: y2+2y−8=(y+4)(y−2)y^2 + 2y - 8 = (y + 4)(y - 2). The second denominator, 2−y2 - y, is the opposite of y−2y - 2. Multiply the second fraction by −1−1\frac{-1}{-1}: y−12−y⋅−1−1=−(y−1)y−2\frac{y - 1}{2 - y} \cdot \frac{-1}{-1} = \frac{-(y - 1)}{y - 2} Rewrite the subtraction: −2y−2(y+4)(y−2)−−(y−1)y−2\frac{-2y - 2}{(y + 4)(y - 2)} - \frac{-(y - 1)}{y - 2}

Step 2 — Simplify the signs. Subtracting a negative is the same as adding a positive (a−(−b)=a+ba - (-b) = a + b): −2y−2(y+4)(y−2)+y−1y−2\frac{-2y - 2}{(y + 4)(y - 2)} + \frac{y - 1}{y - 2}

Step 3 — Find the LCD and rewrite. The LCD is (y+4)(y−2)(y + 4)(y - 2). Multiply the numerator and denominator of the second fraction by (y+4)(y + 4): −2y−2(y+4)(y−2)+(y−1)(y+4)(y+4)(y−2)\frac{-2y - 2}{(y + 4)(y - 2)} + \frac{(y - 1)(y + 4)}{(y + 4)(y - 2)} Expand the second numerator using FOIL: (y−1)(y+4)=y2+3y−4(y - 1)(y + 4) = y^2 + 3y - 4. −2y−2(y+4)(y−2)+y2+3y−4(y+4)(y−2)\frac{-2y - 2}{(y + 4)(y - 2)} + \frac{y^2 + 3y - 4}{(y + 4)(y - 2)}

Step 4 — Add the numerators. Combine over the common denominator: −2y−2+y2+3y−4(y+4)(y−2)\frac{-2y - 2 + y^2 + 3y - 4}{(y + 4)(y - 2)} Combine like terms and rearrange into standard form: y2+y−6(y+4)(y−2)\frac{y^2 + y - 6}{(y + 4)(y - 2)}

Step 5 — Factor and simplify. Factor the new numerator: y2+y−6=(y+3)(y−2)y^2 + y - 6 = (y + 3)(y - 2). (y+3)(y−2)(y+4)(y−2)\frac{(y + 3)(y - 2)}{(y + 4)(y - 2)} Cancel the common factor (y−2)(y - 2) to find the final simplified expression: y+3y+4\frac{y + 3}{y + 4}

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Updated 2026-05-26

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References

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OpenStax

Intermediate Algebra @ OpenStax

Ch.7 Rational Expressions and Functions - Intermediate Algebra @ OpenStax

Algebra

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