Question

directions: on #8-11, divide.

1. \\(\frac{6x - 6}{x} \div \frac{7x - 7}{9x^2}\\)

\\(\frac{6x - 6}{x} \cdot \frac{9x^2}{7x - 7}\\)

1. \\(\frac{x^2 + 5x + 6}{x^2 + 10x + 21} \div \frac{x^2 + 2x}{x^2 + 16x + 63}\\)
2. \\(\frac{(x - 11)^2}{2} \div \frac{2x - 22}{4}\\)
3. \\(\frac{x^2 - 16x + 60}{x - 10} \div (x - 6)\\)

Explanation:

Problem 8

Step1: Rewrite division as multiplication

To divide two fractions, we multiply the first fraction by the reciprocal of the second fraction. So, \(\frac{6x - 6}{x}\div\frac{7x - 7}{9x^{2}}=\frac{6x - 6}{x}\times\frac{9x^{2}}{7x - 7}\)

Step2: Factor numerators and denominators

Factor out the common factors from the numerators and denominators.
For \(6x - 6\), we can factor out 6: \(6x - 6=6(x - 1)\)
For \(7x - 7\), we can factor out 7: \(7x - 7 = 7(x - 1)\)
So the expression becomes \(\frac{6(x - 1)}{x}\times\frac{9x^{2}}{7(x - 1)}\)

Step3: Cancel out common factors

We can cancel out the common factors \((x - 1)\) and one \(x\) from the numerator and the denominator.
\(\frac{6\cancel{(x - 1)}}{\cancel{x}}\times\frac{9x^{\cancel{2}}}{7\cancel{(x - 1)}}=\frac{6\times9x}{7}\)

Step4: Multiply the remaining terms

Multiply 6 and 9: \(6\times9 = 54\)
So the result is \(\frac{54x}{7}\)

Step1: Rewrite division as multiplication

\(\frac{x^{2}+5x + 6}{x^{2}+10x + 21}\div\frac{x^{2}+2x}{x^{2}+16x + 63}=\frac{x^{2}+5x + 6}{x^{2}+10x + 21}\times\frac{x^{2}+16x + 63}{x^{2}+2x}\)

Step2: Factor all quadratics

Factor \(x^{2}+5x + 6\): We need two numbers that multiply to 6 and add to 5. The numbers are 2 and 3. So \(x^{2}+5x + 6=(x + 2)(x + 3)\)
Factor \(x^{2}+10x + 21\): We need two numbers that multiply to 21 and add to 10. The numbers are 3 and 7. So \(x^{2}+10x + 21=(x + 3)(x + 7)\)
Factor \(x^{2}+16x + 63\): We need two numbers that multiply to 63 and add to 16. The numbers are 7 and 9. So \(x^{2}+16x + 63=(x + 7)(x + 9)\)
Factor \(x^{2}+2x\): Factor out \(x\), so \(x^{2}+2x=x(x + 2)\)
Substituting these factorizations into the expression, we get \(\frac{(x + 2)(x + 3)}{(x + 3)(x + 7)}\times\frac{(x + 7)(x + 9)}{x(x + 2)}\)

Step3: Cancel out common factors

Cancel out \((x + 2)\), \((x + 3)\) and \((x + 7)\) from the numerator and the denominator.
\(\frac{\cancel{(x + 2)}\cancel{(x + 3)}}{\cancel{(x + 3)}\cancel{(x + 7)}}\times\frac{\cancel{(x + 7)}(x + 9)}{x\cancel{(x + 2)}}=\frac{x + 9}{x}\)

Step1: Rewrite division as multiplication

\(\frac{(x - 11)^{2}}{2}\div\frac{2x - 22}{4}=\frac{(x - 11)^{2}}{2}\times\frac{4}{2x - 22}\)

Step2: Factor the denominator of the second fraction

Factor out 2 from \(2x - 22\): \(2x - 22 = 2(x - 11)\)
So the expression becomes \(\frac{(x - 11)^{2}}{2}\times\frac{4}{2(x - 11)}\)

Step3: Simplify the constants and cancel common factors

Simplify \(\frac{4}{2}=2\). Then cancel out one \((x - 11)\) from the numerator and the denominator.
\(\frac{(x - 11)^{\cancel{2}}}{2}\times\frac{2}{\cancel{2(x - 11)}}=\frac{(x - 11)\times2}{2}\)

Step4: Cancel out the common factor of 2

Cancel out the 2 in the numerator and the denominator.
\(\frac{(x - 11)\cancel{2}}{\cancel{2}}=x - 11\)

Answer:

\(\frac{54x}{7}\)

Problem 9