Certainly! Let’s work through the division step by step.
1. Identify the given expressions:
- The numerator of the first fraction is [tex]\( x + 3 \)[/tex].
- The denominator of the first fraction is [tex]\( 7 \)[/tex].
- The numerator of the second fraction is [tex]\( 5x + 15 \)[/tex].
- The denominator of the second fraction is [tex]\( 6 \)[/tex].
So, we have:
[tex]\[
\frac{x+3}{7} \quad \text{and} \quad \frac{5x+15}{6}
\][/tex]
2. Rewrite the division as multiplication by the reciprocal:
- When you divide by a fraction, you multiply by its reciprocal. Therefore:
[tex]\[
\frac{x+3}{7} \div \frac{5x+15}{6} = \frac{x+3}{7} \times \frac{6}{5x+15}
\][/tex]
3. Simplify the multiplication:
- Multiply the numerators together:
[tex]\[
(x + 3) \times 6 = 6(x + 3)
\][/tex]
- Multiply the denominators together:
[tex]\[
7 \times (5x + 15) = 7(5x + 15)
\][/tex]
- Therefore:
[tex]\[
\frac{x+3}{7} \times \frac{6}{5x+15} = \frac{6(x + 3)}{7(5x + 15)}
\][/tex]
4. Simplify the resulting expression:
- First, notice that the numerator can be simplified:
[tex]\[
6(x + 3) = 6x + 18
\][/tex]
- The denominator can also be rewritten:
[tex]\[
7(5x + 15) = 35x + 105
\][/tex]
- Now we have:
[tex]\[
\frac{6(x + 3)}{7(5x + 15)} = \frac{6x + 18}{35x + 105}
\][/tex]
5. Further simplify:
- Factor common terms out of both the numerator and the denominator:
- Numerator: [tex]\( 6x + 18 = 6(x + 3) \)[/tex]
- Denominator: [tex]\( 35x + 105 = 35(x + 3) \)[/tex]
- Thus, the expression becomes:
[tex]\[
\frac{6(x + 3)}{35(x + 3)}
\][/tex]
- Since [tex]\( (x + 3) \)[/tex] is a common factor in both the numerator and the denominator, it cancels out:
[tex]\[
\frac{6}{35}
\][/tex]
6. Final result:
- After canceling out the common factor, the expression simplifies to:
[tex]\[
\frac{6}{35}
\][/tex]
Therefore, the simplified result of dividing [tex]\(\frac{x+3}{7}\)[/tex] by [tex]\(\frac{5x+15}{6}\)[/tex] is [tex]\(\frac{6}{35}\)[/tex].
