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(3) \( \frac{4}{3 x}-\frac{x+4}{x-2} \)

Ask by Bradley Estrada. in South Africa
Feb 20,2025

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The simplified form of the expression is \( -\frac{8x+8+3x^2}{3x^2-6x} \).

Solución

Calculate or simplify the expression \( 4/(3*x)-((x+4)/(x-2)) \). Simplify the expression by following steps: - step0: Solution: \(\frac{4}{3x}-\left(\frac{\left(x+4\right)}{\left(x-2\right)}\right)\) - step1: Remove the parentheses: \(\frac{4}{3x}-\left(\frac{x+4}{x-2}\right)\) - step2: Remove the parentheses: \(\frac{4}{3x}-\frac{x+4}{x-2}\) - step3: Reduce fractions to a common denominator: \(\frac{4\left(x-2\right)}{3x\left(x-2\right)}-\frac{\left(x+4\right)\times 3x}{\left(x-2\right)\times 3x}\) - step4: Reorder the terms: \(\frac{4\left(x-2\right)}{3x\left(x-2\right)}-\frac{\left(x+4\right)\times 3x}{3\left(x-2\right)x}\) - step5: Rewrite the expression: \(\frac{4\left(x-2\right)}{3x\left(x-2\right)}-\frac{\left(x+4\right)\times 3x}{3x\left(x-2\right)}\) - step6: Transform the expression: \(\frac{4\left(x-2\right)-\left(x+4\right)\times 3x}{3x\left(x-2\right)}\) - step7: Multiply the terms: \(\frac{4x-8-\left(x+4\right)\times 3x}{3x\left(x-2\right)}\) - step8: Multiply the terms: \(\frac{4x-8-\left(3x^{2}+12x\right)}{3x\left(x-2\right)}\) - step9: Subtract the terms: \(\frac{-8x-8-3x^{2}}{3x\left(x-2\right)}\) - step10: Rewrite the fraction: \(-\frac{8x+8+3x^{2}}{3x\left(x-2\right)}\) - step11: Multiply the terms: \(-\frac{8x+8+3x^{2}}{3x^{2}-6x}\) The simplified form of the expression \( \frac{4}{3x} - \frac{x+4}{x-2} \) is \( -\frac{8x+8+3x^2}{3x^2-6x} \).

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Let's simplify the expression \( \frac{4}{3x} - \frac{x+4}{x-2} \) step by step. First, we need a common denominator to combine these fractions. The denominators are \(3x\) and \(x-2\). The least common denominator (LCD) will be \(3x(x-2)\). Now, we will rewrite each fraction with the common denominator: 1. For the first term \( \frac{4}{3x} \): \[ \frac{4}{3x} = \frac{4(x-2)}{3x(x-2)} = \frac{4x - 8}{3x(x-2)} \] 2. For the second term \( \frac{x+4}{x-2} \): \[ \frac{x+4}{x-2} = \frac{(x+4)(3x)}{(x-2)(3x)} = \frac{3x^2 + 12x}{3x(x-2)} \] Now we can combine the two fractions: \[ \frac{4x - 8}{3x(x-2)} - \frac{3x^2 + 12x}{3x(x-2)} = \frac{(4x - 8) - (3x^2 + 12x)}{3x(x-2)} \] \[ = \frac{4x - 8 - 3x^2 - 12x}{3x(x-2)} = \frac{-3x^2 - 8x - 8}{3x(x-2)} \] Now we can factor the numerator if possible. Let's factor out \(-1\): \[ = \frac{-1(3x^2 + 8x + 8)}{3x(x-2)} \] Finally, since \(3x^2 + 8x + 8\) does not factor nicely (the discriminant is negative), we can leave the expression in this form or further simplify it if needed. Thus, the simplified expression is: \[ \frac{-3x^2 - 8x - 8}{3x(x-2)} \]

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