Question

Simplify the following expression. (1/2x + 5/6) + (4/9x + 5/7)

Answer 1

To simplify the expression \((1/2x + 5/6) + (4/9x + 5/7)\), we first combine the like terms:

1. Combine the \(x\) terms:
\[
\frac{1}{2}x + \frac{4}{9}x
\]

To add these fractions, we need a common denominator. The least common multiple of 2 and 9 is 18.

Rewrite \(\frac{1}{2}x\) and \(\frac{4}{9}x\) with a denominator of 18:
\[
\frac{1}{2}x = \frac{1 \cdot 9}{2 \cdot 9}x = \frac{9}{18}x
\]
\[
\frac{4}{9}x = \frac{4 \cdot 2}{9 \cdot 2}x = \frac{8}{18}x
\]

Now, add them together:
\[
\frac{9}{18}x + \frac{8}{18}x = \frac{9 + 8}{18}x = \frac{17}{18}x
\]

2. Next, combine the constant terms:
\[
\frac{5}{6} + \frac{5}{7}
\]

Again, we need a common denominator, which is 42.

Rewrite \(\frac{5}{6}\) and \(\frac{5}{7}\) with a denominator of 42:
\[
\frac{5}{6} = \frac{5 \cdot 7}{6 \cdot 7} = \frac{35}{42}
\]
\[
\frac{5}{7} = \frac{5 \cdot 6}{7 \cdot 6} = \frac{30}{42}
\]

Now, add them together:
\[
\frac{35}{42} + \frac{30}{42} = \frac{35 + 30}{42} = \frac{65}{42}
\]

Putting it all together, the simplified expression is:
\[
\frac{17}{18}x + \frac{65}{42}
\]

Thus, the final answer is:
\[
\frac{17}{18}x + \frac{65}{42}
\]