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QuestionSimplify these expressions: 1) x2−3x−4x2+x÷x−4x2\frac{x^{2}-3 x-4}{x^{2}+x} \div \frac{x-4}{x^{2}} 2) x−1x+2+x+12x+4\frac{x-1}{x+2}+\frac{x+1}{2 x+4}

Studdy Solution

STEP 1

Assumptions1. We are dealing with algebraic expressions. . The operations involved are division and addition.
3. The variables in the expressions are real numbers.
Problem.1.1

STEP 2

The first step is to rewrite the division operation as multiplication. We do this by taking the reciprocal of the divisor.
x2−x−4x2+x÷x−4x2=x2−x−4x2+x×x2x−4\frac{x^{2}- x-4}{x^{2}+x} \div \frac{x-4}{x^{2}} = \frac{x^{2}- x-4}{x^{2}+x} \times \frac{x^{2}}{x-4}

STEP 3

Next, we factorize the expressions in the numerators and denominators where possible.
=(x−)(x+1)x(x+1)×x2x−= \frac{(x-)(x+1)}{x(x+1)} \times \frac{x^{2}}{x-}

STEP 4

Now, we cancel out the common factors in the numerator and denominator.
=x2x= \frac{x^{2}}{x}

STEP 5

Finally, we simplify the expression to get the final answer.
=x= xProblem2.1.2

STEP 6

The first step is to find a common denominator for the two fractions. In this case, the common denominator is 2x(x+2)2x(x+2).
x−1x+2+x+12x+4=2x(x−1)2x(x+2)+(x+1)(x+2)2x(x+2)\frac{x-1}{x+2}+\frac{x+1}{2 x+4} = \frac{2x(x-1)}{2x(x+2)}+\frac{(x+1)(x+2)}{2x(x+2)}

STEP 7

Next, we simplify the numerators.
=2x2−2x2x2+4x+x2+3x+22x2+4x= \frac{2x^{2}-2x}{2x^{2}+4x}+\frac{x^{2}+3x+2}{2x^{2}+4x}

STEP 8

Now, we add the two fractions together.
=2x2−2x+x2+3x+22x2+4x= \frac{2x^{2}-2x+x^{2}+3x+2}{2x^{2}+4x}

STEP 9

Finally, we simplify the numerator to get the final answer.
=3x2+x+22x2+4x= \frac{3x^{2}+x+2}{2x^{2}+4x}

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