Question

factoring expressions continued
11 determine which of the following expressions are equivalent. show your work.

• $\frac{1}{6}(x - 3)$
• $\frac{1}{4}x - \frac{3}{5} - \frac{1}{12}x + \frac{1}{10}$
• $\frac{1}{18}x + \frac{1}{9}x - \frac{1}{2}$

(all three expressions are equivalent to $\frac{1}{6}x - \frac{1}{2}$, so they are equivalent.)
12 explain a different method you could use to solve problem 11.

Explanation:

Simplify each expression:

1. $\frac{1}{6}(x - 3) = \frac{1}{6}x - \frac{1}{2}$
2. $\frac{1}{4}x - \frac{3}{5} - \frac{1}{12}x + \frac{1}{10} = (\frac{3}{12}x - \frac{1}{12}x) + (-\frac{6}{10} + \frac{1}{10}) = \frac{1}{6}x - \frac{1}{2}$
3. $\frac{1}{18}x + \frac{1}{9}x - \frac{1}{2} = (\frac{1}{18}x + \frac{2}{18}x) - \frac{1}{2} = \frac{1}{6}x - \frac{1}{2}$

All simplify to $\frac{1}{6}x - \frac{1}{2}$, so they are equivalent.

Choose $x=6$:

1. $\frac{1}{6}(6 - 3) = 0.5$
2. $\frac{1}{4}(6) - \frac{3}{5} - \frac{1}{12}(6) + \frac{1}{10} = 1.5 - 0.6 - 0.5 + 0.1 = 0.5$
3. $\frac{1}{18}(6) + \frac{1}{9}(6) - \frac{1}{2} = \frac{1}{3} + \frac{2}{3} - 0.5 = 0.5$

All give the same result, so they are equivalent. This substitution method verifies equivalence without simplifying algebraically.

Answer:

All three expressions are equivalent.