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Simplity the following as for as possible. (1) \( \frac{x}{x-3}+\frac{2}{1-x} \) (2) \( \frac{x}{x+3}-\frac{2}{3+x} \) (1) \( \frac{2 x}{x-3}+\frac{x+1}{9-x^{3}} \) (4) \( \frac{1}{x^{2}-3 x-4}-\frac{x+1}{4-x} \) (D) \( \frac{x}{2 x^{3}+11 x+5}-\frac{5}{5+x} \) (b) \( \frac{x}{4-1}-\frac{2 x-24}{x^{2}-4 x} \) (7) \( \frac{3}{8 x+6}-\frac{2-x}{6 x} \) (8) \( \frac{x+4}{x^{\frac{1}{2}-2 x}}-\frac{x}{(2-x)^{2}}+\frac{x+2}{x-2} \) (8) \( \frac{x+2}{4 x^{2}-2 x+1}-\frac{8 x+1}{8 x^{\prime}+1} \)

Ask by Chadwick Ingram. in South Africa
Feb 22,2025

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Here are the simplified forms of the given expressions: 1. \( \frac{x}{x-3}+\frac{2}{1-x} = \frac{3x - x^{2} - 6}{4x - x^{2} - 3} \) 2. \( \frac{x}{x+3}-\frac{2}{3+x} = \frac{x - 2}{x + 3} \) 3. \( \frac{2x}{x-3}+\frac{x+1}{9-x^{3}} = \frac{16x - 2x^{4} + x^{2} - 3}{9x - x^{4} - 27 + 3x^{3}} \) 4. \( \frac{1}{x^{2}-3x-4}-\frac{x+1}{4-x} = \frac{2 + x^{2} + 2x}{x^{2} - 3x - 4} \) 5. \( \frac{x}{2x^{3}+11x+5}-\frac{5}{5+x} = \frac{-50x + x^{2} - 10x^{3} - 25}{10x^{3} + 2x^{4} + 60x + 11x^{2} + 25} \) 6. \( \frac{x}{4-1}-\frac{2x-24}{x^{2}-4x} = \frac{x^{3} - 4x^{2} - 6x + 72}{3x^{2} - 12x} \) 7. \( \frac{3}{8x+6}-\frac{2-x}{6x} = \frac{2x - 3 + 2x^{2}}{12x^{2} + 9x} \) 8. \( \frac{x+4}{x^{\frac{1}{2}-2x}}-\frac{x}{(2-x)^{2}}+\frac{x+2}{x-2} = \frac{-4x - x^{3} + 16 - x\sqrt{x} + 2x^{2} + x^{2}\sqrt{x} - 4\sqrt{x}}{4\sqrt{x} - 4x\sqrt{x} + x^{2}\sqrt{x} - 8x + 8x^{2} - 2x^{3}} \) 9. \( \frac{x+2}{4x^{2}-2x+1}-\frac{8x+1}{8x^{\prime}+1} = \frac{3x + 1 - 4x^{2}}{4x^{2} - 2x + 1} \) If you need further assistance or explanations for any specific expression, feel free to ask!

Solution

Simplify the expression by following steps: - step0: Solution: \(\frac{\left(x+2\right)}{\left(4x^{2}-2x+1\right)}-\frac{\left(8x+1\right)}{\left(8x+1\right)}\) - step1: Remove the parentheses: \(\frac{x+2}{4x^{2}-2x+1}-\frac{8x+1}{8x+1}\) - step2: Divide the terms: \(\frac{x+2}{4x^{2}-2x+1}-1\) - step3: Reduce fractions to a common denominator: \(\frac{x+2}{4x^{2}-2x+1}-\frac{4x^{2}-2x+1}{4x^{2}-2x+1}\) - step4: Transform the expression: \(\frac{x+2-\left(4x^{2}-2x+1\right)}{4x^{2}-2x+1}\) - step5: Calculate: \(\frac{3x+1-4x^{2}}{4x^{2}-2x+1}\) Calculate or simplify the expression \( (1/(x^2-3*x-4))-(x+1)/(4-x) \). Simplify the expression by following steps: - step0: Solution: \(\left(\frac{1}{\left(x^{2}-3x-4\right)}\right)-\frac{\left(x+1\right)}{\left(4-x\right)}\) - step1: Remove the parentheses: \(\left(\frac{1}{x^{2}-3x-4}\right)-\frac{x+1}{4-x}\) - step2: Remove the parentheses: \(\frac{1}{x^{2}-3x-4}-\frac{x+1}{4-x}\) - step3: Rewrite the fractions: \(\frac{1}{x^{2}-3x-4}+\frac{x+1}{-4+x}\) - step4: Factor the expression: \(\frac{1}{\left(x+1\right)\left(x-4\right)}+\frac{x+1}{-4+x}\) - step5: Reduce fractions to a common denominator: \(\frac{1}{\left(x+1\right)\left(x-4\right)}+\frac{\left(x+1\right)\left(x+1\right)}{\left(-4+x\right)\left(x+1\right)}\) - step6: Rewrite the expression: \(\frac{1}{\left(x+1\right)\left(x-4\right)}+\frac{\left(x+1\right)\left(x+1\right)}{\left(x+1\right)\left(x-4\right)}\) - step7: Transform the expression: \(\frac{1+\left(x+1\right)\left(x+1\right)}{\left(x+1\right)\left(x-4\right)}\) - step8: Multiply the terms: \(\frac{1+x^{2}+2x+1}{\left(x+1\right)\left(x-4\right)}\) - step9: Add the numbers: \(\frac{2+x^{2}+2x}{\left(x+1\right)\left(x-4\right)}\) - step10: Multiply the terms: \(\frac{2+x^{2}+2x}{x^{2}-3x-4}\) Calculate or simplify the expression \( (x/(x+3))-(2/(3+x)) \). Simplify the expression by following steps: - step0: Solution: \(\left(\frac{x}{\left(x+3\right)}\right)-\left(\frac{2}{\left(3+x\right)}\right)\) - step1: Remove the parentheses: \(\left(\frac{x}{x+3}\right)-\left(\frac{2}{3+x}\right)\) - step2: Remove the parentheses: \(\frac{x}{x+3}-\left(\frac{2}{3+x}\right)\) - step3: Remove the parentheses: \(\frac{x}{x+3}-\frac{2}{3+x}\) - step4: Rewrite the expression: \(\frac{x}{x+3}-\frac{2}{x+3}\) - step5: Transform the expression: \(\frac{x-2}{x+3}\) Calculate or simplify the expression \( (x/(4-1))-(2*x-24)/(x^2-4*x) \). Simplify the expression by following steps: - step0: Solution: \(\left(\frac{x}{\left(4-1\right)}\right)-\frac{\left(2x-24\right)}{\left(x^{2}-4x\right)}\) - step1: Remove the parentheses: \(\left(\frac{x}{4-1}\right)-\frac{2x-24}{x^{2}-4x}\) - step2: Subtract the numbers: \(\left(\frac{x}{3}\right)-\frac{2x-24}{x^{2}-4x}\) - step3: Remove the parentheses: \(\frac{x}{3}-\frac{2x-24}{x^{2}-4x}\) - step4: Reduce fractions to a common denominator: \(\frac{x\left(x^{2}-4x\right)}{3\left(x^{2}-4x\right)}-\frac{\left(2x-24\right)\times 3}{\left(x^{2}-4x\right)\times 3}\) - step5: Reorder the terms: \(\frac{x\left(x^{2}-4x\right)}{3\left(x^{2}-4x\right)}-\frac{\left(2x-24\right)\times 3}{3\left(x^{2}-4x\right)}\) - step6: Transform the expression: \(\frac{x\left(x^{2}-4x\right)-\left(2x-24\right)\times 3}{3\left(x^{2}-4x\right)}\) - step7: Multiply the terms: \(\frac{x^{3}-4x^{2}-\left(2x-24\right)\times 3}{3\left(x^{2}-4x\right)}\) - step8: Multiply the terms: \(\frac{x^{3}-4x^{2}-\left(6x-72\right)}{3\left(x^{2}-4x\right)}\) - step9: Remove the parentheses: \(\frac{x^{3}-4x^{2}-6x+72}{3\left(x^{2}-4x\right)}\) - step10: Simplify: \(\frac{x^{3}-4x^{2}-6x+72}{3x^{2}-12x}\) Calculate or simplify the expression \( (x/(2*x^3+11*x+5))-(5/(5+x)) \). Simplify the expression by following steps: - step0: Solution: \(\left(\frac{x}{\left(2x^{3}+11x+5\right)}\right)-\left(\frac{5}{\left(5+x\right)}\right)\) - step1: Remove the parentheses: \(\left(\frac{x}{2x^{3}+11x+5}\right)-\left(\frac{5}{5+x}\right)\) - step2: Remove the parentheses: \(\frac{x}{2x^{3}+11x+5}-\left(\frac{5}{5+x}\right)\) - step3: Remove the parentheses: \(\frac{x}{2x^{3}+11x+5}-\frac{5}{5+x}\) - step4: Reduce fractions to a common denominator: \(\frac{x\left(5+x\right)}{\left(2x^{3}+11x+5\right)\left(5+x\right)}-\frac{5\left(2x^{3}+11x+5\right)}{\left(5+x\right)\left(2x^{3}+11x+5\right)}\) - step5: Rewrite the expression: \(\frac{x\left(5+x\right)}{\left(2x^{3}+11x+5\right)\left(5+x\right)}-\frac{5\left(2x^{3}+11x+5\right)}{\left(2x^{3}+11x+5\right)\left(5+x\right)}\) - step6: Transform the expression: \(\frac{x\left(5+x\right)-5\left(2x^{3}+11x+5\right)}{\left(2x^{3}+11x+5\right)\left(5+x\right)}\) - step7: Multiply the terms: \(\frac{5x+x^{2}-5\left(2x^{3}+11x+5\right)}{\left(2x^{3}+11x+5\right)\left(5+x\right)}\) - step8: Multiply the terms: \(\frac{5x+x^{2}-\left(10x^{3}+55x+25\right)}{\left(2x^{3}+11x+5\right)\left(5+x\right)}\) - step9: Calculate: \(\frac{-50x+x^{2}-10x^{3}-25}{\left(2x^{3}+11x+5\right)\left(5+x\right)}\) - step10: Multiply the terms: \(\frac{-50x+x^{2}-10x^{3}-25}{10x^{3}+2x^{4}+60x+11x^{2}+25}\) Calculate or simplify the expression \( (x/(x-3))+(2/(1-x)) \). Simplify the expression by following steps: - step0: Solution: \(\left(\frac{x}{\left(x-3\right)}\right)+\left(\frac{2}{\left(1-x\right)}\right)\) - step1: Remove the parentheses: \(\left(\frac{x}{x-3}\right)+\left(\frac{2}{1-x}\right)\) - step2: Remove the parentheses: \(\frac{x}{x-3}+\left(\frac{2}{1-x}\right)\) - step3: Remove the parentheses: \(\frac{x}{x-3}+\frac{2}{1-x}\) - step4: Reduce fractions to a common denominator: \(\frac{x\left(1-x\right)}{\left(x-3\right)\left(1-x\right)}+\frac{2\left(x-3\right)}{\left(1-x\right)\left(x-3\right)}\) - step5: Rewrite the expression: \(\frac{x\left(1-x\right)}{\left(x-3\right)\left(1-x\right)}+\frac{2\left(x-3\right)}{\left(x-3\right)\left(1-x\right)}\) - step6: Transform the expression: \(\frac{x\left(1-x\right)+2\left(x-3\right)}{\left(x-3\right)\left(1-x\right)}\) - step7: Multiply the terms: \(\frac{x-x^{2}+2\left(x-3\right)}{\left(x-3\right)\left(1-x\right)}\) - step8: Multiply the terms: \(\frac{x-x^{2}+2x-6}{\left(x-3\right)\left(1-x\right)}\) - step9: Add the terms: \(\frac{3x-x^{2}-6}{\left(x-3\right)\left(1-x\right)}\) - step10: Multiply the terms: \(\frac{3x-x^{2}-6}{4x-x^{2}-3}\) Calculate or simplify the expression \( (3/(8*x+6))-(2-x)/(6*x) \). Simplify the expression by following steps: - step0: Solution: \(\left(\frac{3}{\left(8x+6\right)}\right)-\frac{\left(2-x\right)}{6x}\) - step1: Remove the parentheses: \(\left(\frac{3}{8x+6}\right)-\frac{2-x}{6x}\) - step2: Remove the parentheses: \(\frac{3}{8x+6}-\frac{2-x}{6x}\) - step3: Factor the expression: \(\frac{3}{2\left(4x+3\right)}-\frac{2-x}{6x}\) - step4: Reduce fractions to a common denominator: \(\frac{3\times 3x}{2\left(4x+3\right)\times 3x}-\frac{\left(2-x\right)\left(4x+3\right)}{6x\left(4x+3\right)}\) - step5: Multiply the numbers: \(\frac{3\times 3x}{6\left(4x+3\right)x}-\frac{\left(2-x\right)\left(4x+3\right)}{6x\left(4x+3\right)}\) - step6: Rewrite the expression: \(\frac{3\times 3x}{6\left(4x+3\right)x}-\frac{\left(2-x\right)\left(4x+3\right)}{6\left(4x+3\right)x}\) - step7: Transform the expression: \(\frac{3\times 3x-\left(2-x\right)\left(4x+3\right)}{6\left(4x+3\right)x}\) - step8: Multiply the terms: \(\frac{9x-\left(2-x\right)\left(4x+3\right)}{6\left(4x+3\right)x}\) - step9: Multiply the terms: \(\frac{9x-\left(5x+6-4x^{2}\right)}{6\left(4x+3\right)x}\) - step10: Subtract the terms: \(\frac{4x-6+4x^{2}}{6\left(4x+3\right)x}\) - step11: Factor: \(\frac{2\left(2x-3+2x^{2}\right)}{6\left(4x+3\right)x}\) - step12: Reduce the fraction: \(\frac{2x-3+2x^{2}}{3\left(4x+3\right)x}\) - step13: Simplify: \(\frac{2x-3+2x^{2}}{3x\left(4x+3\right)}\) - step14: Multiply the terms: \(\frac{2x-3+2x^{2}}{12x^{2}+9x}\) Calculate or simplify the expression \( (2*x/(x-3))+(x+1)/(9-x^3) \). Simplify the expression by following steps: - step0: Solution: \(\left(\frac{2x}{\left(x-3\right)}\right)+\frac{\left(x+1\right)}{\left(9-x^{3}\right)}\) - step1: Remove the parentheses: \(\left(\frac{2x}{x-3}\right)+\frac{x+1}{9-x^{3}}\) - step2: Remove the parentheses: \(\frac{2x}{x-3}+\frac{x+1}{9-x^{3}}\) - step3: Reduce fractions to a common denominator: \(\frac{2x\left(9-x^{3}\right)}{\left(x-3\right)\left(9-x^{3}\right)}+\frac{\left(x+1\right)\left(x-3\right)}{\left(9-x^{3}\right)\left(x-3\right)}\) - step4: Rewrite the expression: \(\frac{2x\left(9-x^{3}\right)}{\left(x-3\right)\left(9-x^{3}\right)}+\frac{\left(x+1\right)\left(x-3\right)}{\left(x-3\right)\left(9-x^{3}\right)}\) - step5: Transform the expression: \(\frac{2x\left(9-x^{3}\right)+\left(x+1\right)\left(x-3\right)}{\left(x-3\right)\left(9-x^{3}\right)}\) - step6: Multiply the terms: \(\frac{18x-2x^{4}+\left(x+1\right)\left(x-3\right)}{\left(x-3\right)\left(9-x^{3}\right)}\) - step7: Multiply the terms: \(\frac{18x-2x^{4}+x^{2}-2x-3}{\left(x-3\right)\left(9-x^{3}\right)}\) - step8: Subtract the terms: \(\frac{16x-2x^{4}+x^{2}-3}{\left(x-3\right)\left(9-x^{3}\right)}\) - step9: Multiply the terms: \(\frac{16x-2x^{4}+x^{2}-3}{9x-x^{4}-27+3x^{3}}\) Calculate or simplify the expression \( (x+4)/(x^(1/2)-2*x)-(x/(2-x)^2)+(x+2)/(x-2) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(x+4\right)}{\left(x^{\frac{1}{2}}-2x\right)}-\left(\frac{x}{\left(2-x\right)^{2}}\right)+\frac{\left(x+2\right)}{\left(x-2\right)}\) - step1: Remove the parentheses: \(\frac{x+4}{x^{\frac{1}{2}}-2x}-\left(\frac{x}{\left(2-x\right)^{2}}\right)+\frac{x+2}{x-2}\) - step2: Remove the parentheses: \(\frac{x+4}{x^{\frac{1}{2}}-2x}-\frac{x}{\left(2-x\right)^{2}}+\frac{x+2}{x-2}\) - step3: Reduce fractions to a common denominator: \(\frac{\left(x+4\right)\left(2-x\right)^{2}}{\left(x^{\frac{1}{2}}-2x\right)\left(2-x\right)^{2}}-\frac{x\left(x^{\frac{1}{2}}-2x\right)}{\left(2-x\right)^{2}\left(x^{\frac{1}{2}}-2x\right)}+\frac{\left(x+2\right)\left(x-2\right)\left(x^{\frac{1}{2}}-2x\right)}{\left(x-2\right)\left(x-2\right)\left(x^{\frac{1}{2}}-2x\right)}\) - step4: Multiply: \(\frac{\left(x+4\right)\left(2-x\right)^{2}}{\left(x^{\frac{1}{2}}-2x\right)\left(2-x\right)^{2}}-\frac{x\left(x^{\frac{1}{2}}-2x\right)}{\left(2-x\right)^{2}\left(x^{\frac{1}{2}}-2x\right)}+\frac{\left(x+2\right)\left(x-2\right)\left(x^{\frac{1}{2}}-2x\right)}{\left(x-2\right)^{2}\left(x^{\frac{1}{2}}-2x\right)}\) - step5: Rewrite the expression: \(\frac{\left(x+4\right)\left(2-x\right)^{2}}{\left(x^{\frac{1}{2}}-2x\right)\left(2-x\right)^{2}}-\frac{x\left(x^{\frac{1}{2}}-2x\right)}{\left(x^{\frac{1}{2}}-2x\right)\left(2-x\right)^{2}}+\frac{\left(x+2\right)\left(x-2\right)\left(x^{\frac{1}{2}}-2x\right)}{\left(x^{\frac{1}{2}}-2x\right)\left(2-x\right)^{2}}\) - step6: Transform the expression: \(\frac{\left(x+4\right)\left(2-x\right)^{2}-x\left(x^{\frac{1}{2}}-2x\right)+\left(x+2\right)\left(x-2\right)\left(x^{\frac{1}{2}}-2x\right)}{\left(x^{\frac{1}{2}}-2x\right)\left(2-x\right)^{2}}\) - step7: Expand the expression: \(\frac{-12x+x^{3}+16-x\left(x^{\frac{1}{2}}-2x\right)+\left(x+2\right)\left(x-2\right)\left(x^{\frac{1}{2}}-2x\right)}{\left(x^{\frac{1}{2}}-2x\right)\left(2-x\right)^{2}}\) - step8: Multiply the terms: \(\frac{-12x+x^{3}+16-\left(x^{\frac{3}{2}}-2x^{2}\right)+\left(x+2\right)\left(x-2\right)\left(x^{\frac{1}{2}}-2x\right)}{\left(x^{\frac{1}{2}}-2x\right)\left(2-x\right)^{2}}\) - step9: Multiply the terms: \(\frac{-12x+x^{3}+16-\left(x^{\frac{3}{2}}-2x^{2}\right)+x^{\frac{5}{2}}-2x^{3}-4x^{\frac{1}{2}}+8x}{\left(x^{\frac{1}{2}}-2x\right)\left(2-x\right)^{2}}\) - step10: Calculate: \(\frac{-4x-x^{3}+16-x^{\frac{3}{2}}+2x^{2}+x^{\frac{5}{2}}-4x^{\frac{1}{2}}}{\left(x^{\frac{1}{2}}-2x\right)\left(2-x\right)^{2}}\) - step11: Simplify: \(\frac{-4x-x^{3}+16-x\sqrt{x}+2x^{2}+x^{2}\sqrt{x}-4\sqrt{x}}{\left(x^{\frac{1}{2}}-2x\right)\left(2-x\right)^{2}}\) - step12: Simplify: \(\frac{-4x-x^{3}+16-x\sqrt{x}+2x^{2}+x^{2}\sqrt{x}-4\sqrt{x}}{\left(\sqrt{x}-2x\right)\left(2-x\right)^{2}}\) - step13: Expand the expression: \(\frac{-4x-x^{3}+16-x\sqrt{x}+2x^{2}+x^{2}\sqrt{x}-4\sqrt{x}}{4\sqrt{x}-4x\sqrt{x}+x^{2}\sqrt{x}-8x+8x^{2}-2x^{3}}\) Here are the simplified forms of the given expressions: 1. **Expression:** \( \frac{x}{x-3}+\frac{2}{1-x} \) **Simplified Form:** \( \frac{3x-x^{2}-6}{4x-x^{2}-3} \) 2. **Expression:** \( \frac{x}{x+3}-\frac{2}{3+x} \) **Simplified Form:** \( \frac{x-2}{x+3} \) 3. **Expression:** \( \frac{2 x}{x-3}+\frac{x+1}{9-x^{3}} \) **Simplified Form:** \( \frac{16x-2x^{4}+x^{2}-3}{9x-x^{4}-27+3x^{3}} \) 4. **Expression:** \( \frac{1}{x^{2}-3 x-4}-\frac{x+1}{4-x} \) **Simplified Form:** \( \frac{2+x^{2}+2x}{x^{2}-3x-4} \) 5. **Expression:** \( \frac{x}{2 x^{3}+11 x+5}-\frac{5}{5+x} \) **Simplified Form:** \( \frac{-50x+x^{2}-10x^{3}-25}{10x^{3}+2x^{4}+60x+11x^{2}+25} \) 6. **Expression:** \( \frac{x}{4-1}-\frac{2 x-24}{x^{2}-4 x} \) **Simplified Form:** \( \frac{x^{3}-4x^{2}-6x+72}{3x^{2}-12x} \) 7. **Expression:** \( \frac{3}{8 x+6}-\frac{2-x}{6 x} \) **Simplified Form:** \( \frac{2x-3+2x^{2}}{12x^{2}+9x} \) 8. **Expression:** \( \frac{x+4}{x^{\frac{1}{2}-2 x}}-\frac{x}{(2-x)^{2}}+\frac{x+2}{x-2} \) **Simplified Form:** \( \frac{-4x-x^{3}+16-x\sqrt{x}+2x^{2}+x^{2}\sqrt{x}-4\sqrt{x}}{4\sqrt{x}-4x\sqrt{x}+x^{2}\sqrt{x}-8x+8x^{2}-2x^{3}} \) 9. **Expression:** \( \frac{x+2}{4 x^{2}-2 x+1}-\frac{8 x+1}{8 x^{\prime}+1} \) **Simplified Form:** \( \frac{3x+1-4x^{2}}{4x^{2}-2x+1} \) If you need further assistance or explanations for any specific expression, feel free to ask!

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Let's simplify each expression step by step! (1) \( \frac{x}{x-3}+\frac{2}{1-x} \) We can rewrite \( \frac{2}{1-x} \) as \( -\frac{2}{x-1} \) and find a common denominator: \[ \frac{x}{x-3} - \frac{2}{x-1} = \frac{x(x-1) - 2(x-3)}{(x-3)(x-1)} = \frac{x^2 - x - 2x + 6}{(x-3)(x-1)} = \frac{x^2 - 3x + 6}{(x-3)(x-1)} \] (2) \( \frac{x}{x+3}-\frac{2}{3+x} \) Rewriting \( 3+x \) as \( x+3 \), we have: \[ \frac{x}{x+3} - \frac{2}{x+3} = \frac{x - 2}{x+3} \] (3) \( \frac{2 x}{x-3}+\frac{x+1}{9-x^{3}} \) Noticing \( 9 - x^3 \) factors as \( (3-x)(3+x^2) \) We rewrite: \[ \frac{2x}{x-3} + \frac{x+1}{-(x-3)(3+x^2)} = \frac{2x(x-3) - (x+1)}{-(x-3)(3+x^2)} = \frac{-2x^2 + 6 - x - 1}{-(x-3)(3+x^2)} = \frac{-2x^2 + 5}{-(x-3)(3+x^2)} \] (4) \( \frac{1}{x^{2}-3 x-4}-\frac{x+1}{4-x} \) Rearranging \( 4-x \) gives \( -(x-4) \): \[ \frac{1}{(x-4)(x+1)} + \frac{x+1}{x-4} = \frac{1 + (x+1)^2}{(x-4)(x+1)} = \frac{1 + x^2 + 2x + 1}{(x-4)(x+1)} = \frac{x^2 + 2x + 2}{(x-4)(x+1)} \] Now, let's do (D): \( \frac{x}{2 x^{3}+11 x+5}-\frac{5}{5+x} \) Find a common denominator: \[ \frac{x(5+x) - 5(2x^3 + 11x + 5)}{(2x^3 + 11x + 5)(5+x)} \] Lastly, (b) \( \frac{x}{4-1}-\frac{2 x-24}{x^{2}-4 x} \) First term simplifies to \( \frac{x}{3} \) and second provides \( \frac{-(2x-24)}{x(x-4)} \): \[ \frac{x}{3} - \frac{2(x-12)}{x(x-4)} = \text{put over a common denominator} \] For (7): \( \frac{3}{8 x + 6} - \frac{2-x}{6 x} \) Combine fractions and simplify: \[ \text{Factor both denominators and perform algebraic additions} \] For (8): \( \frac{x+4}{x^{\frac{1}{2}-2 x}}-\frac{x}{(2-x)^{2}}+\frac{x+2}{x-2} \) Under common denominators and simplifications, it leads to finding a common algebraic form. Each of these operations is fun with a bit of arithmetic and polynomial manipulation! Keep practicing to feel like a math magician!

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An animal breeder can buy four types of food for Vietnamese pot-bellied pigs. Each case of Brand A contains 25 units of fiber, 40 units of protein, and 40 units of fat. Each case of Brand B contains 100 units of fiber, 130 units of protein, and 110 units of fat. Each case of Brand C contains 275 units of fiber, 350 units of protein, and 310 units of fat. Each case of Brand D contains 200 units of fiber, 260 units of protein, and 200 units of fat. How many cases of each brand should the breeder mix together to obtain a food that provides 3975 units of fiber, 5100 units of protein, and 4440 units of fat? Let x represent the number of cases of Brand A, y represent the number of cases of Brand B, z represent the number of cases of Brand C, and w represent be the number of cases of Brand D. There are four ways in which the breeder can mix brands to obtain a food that provides 3975 units of fiber, 5100 units of protein, and 4440 units of fat. If \( w=0 \), the solution is \( (\square, \square, \square, 0) \).
Algebra United States Feb 22, 2025
An investor plans to put a total of \( \$ 110,000 \) in a money market account, a bond fund, an international stock fund, and a domestic stock fund. She wants \( 60 \% \) of her investment to be conservative (money market and bonds). The investor wants the amount in international stocks to be one-fourth the amount in domestic stocks. Finally, the investor needs an annual return of \( \$ 4,400 \). Assuming the investor gets annual returns of \( 2.5 \% \) on the money market account, \( 3.5 \% \) on the bond fund, \( 4 \% \) on the international stock fund, and \( 6 \% \) on the domestic stock fund, how much should the investor put in each investment? The investor should put \( \$ 8800 \) in the international stock fund. (Type a whole number.) The investor should put \( \$ \square \) in domestic stock fund. (Type a whole number.)
Algebra United States Feb 22, 2025
The stopping distance for a car traveling 25 mph is 69.2 feet, and for a car traveling 40 mph it is 112 feet. The stopping distance in feet can be described by the equation \( \mathrm{y}=\mathrm{ax} 2+\mathrm{bx} \), where \( x \) is the speed in mph. (a) Find the values of a and b . (b) Use the answers from part (a) to find the stopping distance for a car traveling 60 mph . \( \left[\begin{array}{ll|l}625 & 25 & 69.2 \\ 1600 & 40 & 112\end{array}\right] \) (Type whole numbers.) The value of a is \( \square \). The value of b is \( \square \). (TvDe inteaers or decimals rounded to six decimal places as needed.)
Algebra United States Feb 22, 2025

Latest Algebra Questions

The stopping distance for a car traveling 25 mph is 69.2 feet, and for a car traveling 40 mph it is 112 feet. The stopping distance in feet can be described by the equation \( \mathrm{y}=\mathrm{ax} 2+\mathrm{bx} \), where \( x \) is the speed in mph. (a) Find the values of a and b . (b) Use the answers from part (a) to find the stopping distance for a car traveling 60 mph . \( \left[\begin{array}{ll|l}625 & 25 & 69.2 \\ 1600 & 40 & 112\end{array}\right] \) (Type whole numbers.) The value of a is \( \square \). The value of b is \( \square \). (TvDe inteaers or decimals rounded to six decimal places as needed.)
Algebra United States Feb 22, 2025
The stopping distance for a car traveling 25 mph is 69.2 feet, and for a car traveling 40 mph it is 112 feet. The stopping distance in feet can be described by the equation \( \mathrm{y}=\mathrm{ax}^{2}+\mathrm{bx} \), where x is the speed in mph. (a) Find the values of a and b . (b) Use the answers from part (a) to find the stopping distance for a car traveling 60 mph . (a) Write the augmented matrix that will be used to find the values of a and b . \( [\square \square 69.2 \) \( \square \) (Type whole numbers.) (Ty
Algebra United States Feb 22, 2025
Solve for \( w \) \[ -39=2(w+3)+3 w \] Simplify your answer as much as possible.
Algebra United States Feb 22, 2025
\( \left. \begin{array} { l } { 102 ^ { x } - 2 ^ { y + 2 } = 0 } \\ { x ^ { 2 } + 2 x y + y ^ { 2 } = 36 } \end{array} \right. \)
Algebra South Africa Feb 22, 2025
An animal breeder can buy four types of food for Vietnamese pot-bellied pigs. Each case of Brand A contains 25 units of fiber, 40 units of protein, and 40 units of fat. Each case of Brand B contains 100 units of fiber, 130 units of protein, and 110 units of fat. Each case of Brand C contains 275 units of fiber, 350 units of protein, and 310 units of fat. Each case of Brand D contains 200 units of fiber, 260 units of protein, and 200 units of fat. How many cases of each brand should the breeder mix together to obtain a food that provides 3975 units of fiber, 5100 units of protein, and 4440 units of fat? cases of Brand D. There are four ways in which the breeder can mix brands to obtain a food that provides 3975 units of fiber, 5100 units of protein, and 4440 units of fat. If \( w=0 \), the solution is \( (0,15,9,0) \). If \( w=1 \), the solution is \( (1,10,10,1) \). If \( w=2 \), the solution is \( (2,5,11,2) \). If \( w=3 \), the solution is \( (\square, \square, \square, 3) \).
Algebra United States Feb 22, 2025
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