ERCISE 15 Simplify the following as far as possible. \( \begin{array}{ll}\text { (1) } \frac{x}{x-3}+\frac{2}{3-x} & \text { (2) } \frac{x}{x+3}-\frac{2}{3+x} \\ \text { (3) } \frac{2 x}{x-3}+\frac{x+1}{9-x^{2}} & \text { (4) } \frac{3}{x^{2}-3 x-4}-\frac{x+1}{4-x} \\ \text { (5) } \frac{x}{2 x^{2}+11 x+5}-\frac{5}{5+x} & \text { (6) } \frac{x}{4-x}-\frac{2 x-24}{x^{2}-4 x} \\ \text { (7) } \frac{3}{6 x+6}-\frac{2-x}{6 x} & \text { (8) } \frac{x+4}{x^{2}-2 x}-\frac{x}{(2-x)^{2}}+\frac{x+2}{x-2} \\ \text { (9) } \frac{x+2}{4 x^{2}-2 x+1}-\frac{8 x+1}{8 x^{3}+1} & \end{array} \)

Simplify the expression by following steps: - step0: Solution: \(\frac{x}{\left(x-3\right)}+\frac{2}{\left(3-x\right)}\) - step1: Remove the parentheses: \(\frac{x}{x-3}+\frac{2}{3-x}\) - step2: Rewrite the fractions: \(\frac{x}{x-3}-\frac{2}{-3+x}\) - step3: Rewrite the expression: \(\frac{x}{x-3}-\frac{2}{x-3}\) - step4: Transform the expression: \(\frac{x-2}{x-3}\) Calculate or simplify the expression \( x/(x+3) - 2/(3+x) \). Simplify the expression by following steps: - step0: Solution: \(\frac{x}{\left(x+3\right)}-\frac{2}{\left(3+x\right)}\) - step1: Remove the parentheses: \(\frac{x}{x+3}-\frac{2}{3+x}\) - step2: Rewrite the expression: \(\frac{x}{x+3}-\frac{2}{x+3}\) - step3: Transform the expression: \(\frac{x-2}{x+3}\) Calculate or simplify the expression \( x/(2*x^2 + 11*x + 5) - 5/(5+x) \). Simplify the expression by following steps: - step0: Solution: \(\frac{x}{\left(2x^{2}+11x+5\right)}-\frac{5}{\left(5+x\right)}\) - step1: Remove the parentheses: \(\frac{x}{2x^{2}+11x+5}-\frac{5}{5+x}\) - step2: Factor the expression: \(\frac{x}{\left(2x+1\right)\left(x+5\right)}-\frac{5}{5+x}\) - step3: Reduce fractions to a common denominator: \(\frac{x}{\left(2x+1\right)\left(x+5\right)}-\frac{5\left(2x+1\right)}{\left(5+x\right)\left(2x+1\right)}\) - step4: Rewrite the expression: \(\frac{x}{\left(2x+1\right)\left(x+5\right)}-\frac{5\left(2x+1\right)}{\left(2x+1\right)\left(x+5\right)}\) - step5: Transform the expression: \(\frac{x-5\left(2x+1\right)}{\left(2x+1\right)\left(x+5\right)}\) - step6: Multiply the terms: \(\frac{x-\left(10x+5\right)}{\left(2x+1\right)\left(x+5\right)}\) - step7: Subtract the terms: \(\frac{-9x-5}{\left(2x+1\right)\left(x+5\right)}\) - step8: Rewrite the fraction: \(-\frac{9x+5}{\left(2x+1\right)\left(x+5\right)}\) - step9: Multiply the terms: \(-\frac{9x+5}{2x^{2}+11x+5}\) Calculate or simplify the expression \( x/(4-x) - (2*x-24)/(x^2-4*x) \). Simplify the expression by following steps: - step0: Solution: \(\frac{x}{\left(4-x\right)}-\frac{\left(2x-24\right)}{\left(x^{2}-4x\right)}\) - step1: Remove the parentheses: \(\frac{x}{4-x}-\frac{2x-24}{x^{2}-4x}\) - step2: Factor the expression: \(\frac{x}{4-x}-\frac{2x-24}{x\left(x-4\right)}\) - step3: Reduce fractions to a common denominator: \(\frac{x\times x}{\left(4-x\right)x}-\frac{\left(2x-24\right)\left(-1\right)}{x\left(x-4\right)\left(-1\right)}\) - step4: Reorder the terms: \(\frac{x\times x}{\left(4-x\right)x}-\frac{\left(2x-24\right)\left(-1\right)}{-x\left(x-4\right)}\) - step5: Rewrite the expression: \(\frac{x\times x}{\left(4-x\right)x}-\frac{\left(2x-24\right)\left(-1\right)}{\left(4-x\right)x}\) - step6: Transform the expression: \(\frac{x\times x-\left(2x-24\right)\left(-1\right)}{\left(4-x\right)x}\) - step7: Multiply the terms: \(\frac{x^{2}-\left(2x-24\right)\left(-1\right)}{\left(4-x\right)x}\) - step8: Multiply the terms: \(\frac{x^{2}-\left(-2x+24\right)}{\left(4-x\right)x}\) - step9: Remove the parentheses: \(\frac{x^{2}+2x-24}{\left(4-x\right)x}\) - step10: Factor the expression: \(\frac{\left(-x+4\right)\left(-x-6\right)}{\left(-x+4\right)x}\) - step11: Reduce the fraction: \(\frac{-x-6}{x}\) - step12: Calculate: \(-\frac{x+6}{x}\) Calculate or simplify the expression \( (x+2)/(4*x^2-2*x+1) - (8*x+1)/(8*x^3+1) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(x+2\right)}{\left(4x^{2}-2x+1\right)}-\frac{\left(8x+1\right)}{\left(8x^{3}+1\right)}\) - step1: Remove the parentheses: \(\frac{x+2}{4x^{2}-2x+1}-\frac{8x+1}{8x^{3}+1}\) - step2: Factor the expression: \(\frac{x+2}{4x^{2}-2x+1}-\frac{8x+1}{\left(2x+1\right)\left(4x^{2}-2x+1\right)}\) - step3: Reduce fractions to a common denominator: \(\frac{\left(x+2\right)\left(2x+1\right)}{\left(4x^{2}-2x+1\right)\left(2x+1\right)}-\frac{8x+1}{\left(2x+1\right)\left(4x^{2}-2x+1\right)}\) - step4: Rewrite the expression: \(\frac{\left(x+2\right)\left(2x+1\right)}{\left(2x+1\right)\left(4x^{2}-2x+1\right)}-\frac{8x+1}{\left(2x+1\right)\left(4x^{2}-2x+1\right)}\) - step5: Transform the expression: \(\frac{\left(x+2\right)\left(2x+1\right)-\left(8x+1\right)}{\left(2x+1\right)\left(4x^{2}-2x+1\right)}\) - step6: Multiply the terms: \(\frac{2x^{2}+5x+2-\left(8x+1\right)}{\left(2x+1\right)\left(4x^{2}-2x+1\right)}\) - step7: Calculate: \(\frac{2x^{2}-3x+1}{\left(2x+1\right)\left(4x^{2}-2x+1\right)}\) - step8: Simplify the product: \(\frac{2x^{2}-3x+1}{8x^{3}+1}\) Calculate or simplify the expression \( 3/(6*x+6) - (2-x)/(6*x) \). Simplify the expression by following steps: - step0: Solution: \(\frac{3}{\left(6x+6\right)}-\frac{\left(2-x\right)}{6x}\) - step1: Remove the parentheses: \(\frac{3}{6x+6}-\frac{2-x}{6x}\) - step2: Divide the terms: \(\frac{1}{2x+2}-\frac{2-x}{6x}\) - step3: Factor the expression: \(\frac{1}{2\left(x+1\right)}-\frac{2-x}{6x}\) - step4: Reduce fractions to a common denominator: \(\frac{3x}{2\left(x+1\right)\times 3x}-\frac{\left(2-x\right)\left(x+1\right)}{6x\left(x+1\right)}\) - step5: Multiply the numbers: \(\frac{3x}{6\left(x+1\right)x}-\frac{\left(2-x\right)\left(x+1\right)}{6x\left(x+1\right)}\) - step6: Rewrite the expression: \(\frac{3x}{6\left(x+1\right)x}-\frac{\left(2-x\right)\left(x+1\right)}{6\left(x+1\right)x}\) - step7: Transform the expression: \(\frac{3x-\left(2-x\right)\left(x+1\right)}{6\left(x+1\right)x}\) - step8: Multiply the terms: \(\frac{3x-\left(x+2-x^{2}\right)}{6\left(x+1\right)x}\) - step9: Subtract the terms: \(\frac{2x-2+x^{2}}{6\left(x+1\right)x}\) - step10: Calculate: \(\frac{2x-2+x^{2}}{6x^{2}+6x}\) Calculate or simplify the expression \( 2*x/(x-3) + (x+1)/(9-x^2) \). Simplify the expression by following steps: - step0: Solution: \(\frac{2x}{\left(x-3\right)}+\frac{\left(x+1\right)}{\left(9-x^{2}\right)}\) - step1: Remove the parentheses: \(\frac{2x}{x-3}+\frac{x+1}{9-x^{2}}\) - step2: Factor the expression: \(\frac{2x}{x-3}+\frac{x+1}{\left(-3-x\right)\left(x-3\right)}\) - step3: Reduce fractions to a common denominator: \(\frac{2x\left(-3-x\right)}{\left(x-3\right)\left(-3-x\right)}+\frac{x+1}{\left(-3-x\right)\left(x-3\right)}\) - step4: Rewrite the expression: \(\frac{2x\left(-3-x\right)}{\left(-3-x\right)\left(x-3\right)}+\frac{x+1}{\left(-3-x\right)\left(x-3\right)}\) - step5: Transform the expression: \(\frac{2x\left(-3-x\right)+x+1}{\left(-3-x\right)\left(x-3\right)}\) - step6: Multiply the terms: \(\frac{-6x-2x^{2}+x+1}{\left(-3-x\right)\left(x-3\right)}\) - step7: Add the terms: \(\frac{-5x-2x^{2}+1}{\left(-3-x\right)\left(x-3\right)}\) - step8: Rewrite the fraction: \(\frac{5x+2x^{2}-1}{\left(3+x\right)\left(x-3\right)}\) - step9: Multiply the terms: \(\frac{5x+2x^{2}-1}{x^{2}-9}\) Calculate or simplify the expression \( 3/(x^2-3*x-4) - (x+1)/(4-x) \). Simplify the expression by following steps: - step0: Solution: \(\frac{3}{\left(x^{2}-3x-4\right)}-\frac{\left(x+1\right)}{\left(4-x\right)}\) - step1: Remove the parentheses: \(\frac{3}{x^{2}-3x-4}-\frac{x+1}{4-x}\) - step2: Rewrite the fractions: \(\frac{3}{x^{2}-3x-4}+\frac{x+1}{-4+x}\) - step3: Factor the expression: \(\frac{3}{\left(x+1\right)\left(x-4\right)}+\frac{x+1}{-4+x}\) - step4: Reduce fractions to a common denominator: \(\frac{3}{\left(x+1\right)\left(x-4\right)}+\frac{\left(x+1\right)\left(x+1\right)}{\left(-4+x\right)\left(x+1\right)}\) - step5: Rewrite the expression: \(\frac{3}{\left(x+1\right)\left(x-4\right)}+\frac{\left(x+1\right)\left(x+1\right)}{\left(x+1\right)\left(x-4\right)}\) - step6: Transform the expression: \(\frac{3+\left(x+1\right)\left(x+1\right)}{\left(x+1\right)\left(x-4\right)}\) - step7: Multiply the terms: \(\frac{3+x^{2}+2x+1}{\left(x+1\right)\left(x-4\right)}\) - step8: Add the numbers: \(\frac{4+x^{2}+2x}{\left(x+1\right)\left(x-4\right)}\) - step9: Multiply the terms: \(\frac{4+x^{2}+2x}{x^{2}-3x-4}\) Calculate or simplify the expression \( (x+4)/(x^2-2*x) - x/(2-x)^2 + (x+2)/(x-2) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(x+4\right)}{\left(x^{2}-2x\right)}-\frac{x}{\left(2-x\right)^{2}}+\frac{\left(x+2\right)}{\left(x-2\right)}\) - step1: Remove the parentheses: \(\frac{x+4}{x^{2}-2x}-\frac{x}{\left(2-x\right)^{2}}+\frac{x+2}{x-2}\) - step2: Factor the expression: \(\frac{x+4}{x\left(x-2\right)}-\frac{x}{\left(2-x\right)^{2}}+\frac{x+2}{x-2}\) - step3: Reduce fractions to a common denominator: \(\frac{\left(x+4\right)\left(x-2\right)}{x\left(x-2\right)\left(x-2\right)}-\frac{x\times x}{\left(2-x\right)^{2}x}+\frac{\left(x+2\right)\left(x-2\right)x}{\left(x-2\right)\left(x-2\right)x}\) - step4: Multiply: \(\frac{\left(x+4\right)\left(x-2\right)}{x\left(x-2\right)^{2}}-\frac{x\times x}{\left(2-x\right)^{2}x}+\frac{\left(x+2\right)\left(x-2\right)x}{\left(x-2\right)\left(x-2\right)x}\) - step5: Multiply: \(\frac{\left(x+4\right)\left(x-2\right)}{x\left(x-2\right)^{2}}-\frac{x\times x}{\left(2-x\right)^{2}x}+\frac{\left(x+2\right)\left(x-2\right)x}{\left(x-2\right)^{2}x}\) - step6: Rewrite the expression: \(\frac{\left(x+4\right)\left(x-2\right)}{x\left(x-2\right)^{2}}-\frac{x\times x}{x\left(x-2\right)^{2}}+\frac{\left(x+2\right)\left(x-2\right)x}{x\left(x-2\right)^{2}}\) - step7: Transform the expression: \(\frac{\left(x+4\right)\left(x-2\right)-x\times x+\left(x+2\right)\left(x-2\right)x}{x\left(x-2\right)^{2}}\) - step8: Multiply the terms: \(\frac{x^{2}+2x-8-x\times x+\left(x+2\right)\left(x-2\right)x}{x\left(x-2\right)^{2}}\) - step9: Multiply the terms: \(\frac{x^{2}+2x-8-x^{2}+\left(x+2\right)\left(x-2\right)x}{x\left(x-2\right)^{2}}\) - step10: Multiply the terms: \(\frac{x^{2}+2x-8-x^{2}+x^{3}-4x}{x\left(x-2\right)^{2}}\) - step11: Calculate: \(\frac{-2x-8+x^{3}}{x\left(x-2\right)^{2}}\) - step12: Expand the expression: \(\frac{-2x-8+x^{3}}{x^{3}-4x^{2}+4x}\) Here are the simplified results for each of the expressions: 1. **Expression:** \( \frac{x}{x-3} + \frac{2}{3-x} \) **Simplified Result:** \( \frac{x-2}{x-3} \) 2. **Expression:** \( \frac{x}{x+3} - \frac{2}{3+x} \) **Simplified Result:** \( \frac{x-2}{x+3} \) 3. **Expression:** \( \frac{2x}{x-3} + \frac{x+1}{9-x^2} \) **Simplified Result:** \( \frac{5x + 2x^2 - 1}{x^2 - 9} \) 4. **Expression:** \( \frac{3}{x^2 - 3x - 4} - \frac{x+1}{4-x} \) **Simplified Result:** \( \frac{4 + x^2 + 2x}{x^2 - 3x - 4} \) 5. **Expression:** \( \frac{x}{2x^2 + 11x + 5} - \frac{5}{5+x} \) **Simplified Result:** \( -\frac{9x + 5}{2x^2 + 11x + 5} \) 6. **Expression:** \( \frac{x}{4-x} - \frac{2x-24}{x^2 - 4x} \) **Simplified Result:** \( -\frac{x + 6}{x} \) 7. **Expression:** \( \frac{3}{6x + 6} - \frac{2-x}{6x} \) **Simplified Result:** \( \frac{2x - 2 + x^2}{6x^2 + 6x} \) 8. **Expression:** \( \frac{x+4}{x^2 - 2x} - \frac{x}{(2-x)^2} + \frac{x+2}{x-2} \) **Simplified Result:** \( \frac{-2x - 8 + x^3}{x^3 - 4x^2 + 4x} \) 9. **Expression:** \( \frac{x+2}{4x^2 - 2x + 1} - \frac{8x+1}{8x^3 + 1} \) **Simplified Result:** \( \frac{2x^2 - 3x + 1}{8x^3 + 1} \) These results represent the simplified forms of the given expressions. 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