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Data Sufficiency: Plugging into Yes/No Data Sufficiency

If $$a$$ and $$b$$ are non-zero integers, is $$\frac{1}{a + b} \gt 0$$? >(1) $$a \gt b$$ >(2) $$\frac{1}{a} \gt 0$$
Incorrect. [[snippet]] According to Stat. (1), $$a \gt b$$. However, that doesn't tell you whether $$a$$ and $$b$$ are positive or negative, or one positive–one negative. Plug in values for $$a$$ and $$b$$: * If $$a = -2$$ and $$b = -4$$, then $$\frac{1}{a + b} = -\frac{1}{6}$$, which is _not_ greater than zero. * If $$a = 3$$ and $$b = 2$$, then $$\frac{1}{a + b} = \frac{1}{5}$$, which _is_ greater than zero. No definite answer, so **Stat.(1) → Maybe → IS → BCE**.
Incorrect. [[snippet]] According to Stat. (2), $$\frac{1}{a} \gt 0$$. From here, it follows that $$\frac{1}{a}$$ is a positive fraction, so $$a$$ must be positive as well (otherwise, $$\frac{1}{a}$$ would be negative). However, that doesn't tell you anything about $$b$$. Plug in values for $$a$$ and $$b$$: * If $$a = 3$$ and $$b = 2$$, then $$\frac{1}{a + b} = \frac{1}{5}$$, which _is_ greater than zero. * If $$a = 3$$ and $$b = -7$$, then $$\frac{1}{a + b} = -\frac{1}{4}$$, which is _not_ greater than zero. No definite answer, so **Stat.(2) → Maybe → IS → ACE**.
Incorrect. [[Snippet]] According to Stat. (1+2), $$a \gt b$$ and $$\frac{1}{a} \gt 0$$. So, $$a$$ has to be positive, but $$b$$ can be either positive or negative. Plug in values for $$a$$ and $$b$$: * If $$a = 3$$ and $$b = 2$$, then $$\frac{1}{a + b} = \frac{1}{5}$$, which _is_ greater than zero. * If $$a = 3$$ and $$b = -7$$, then $$\frac{1}{a + b} = -\frac{1}{4}$$, which is _not_ greater than zero. No definite answer, so **Stat.(1+2) → Maybe → IS → E**.
Incorrect. [[snippet]] According to Stat. (1), $$a \gt b$$. However, that doesn't tell you whether $$a$$ and $$b$$ are positive or negative, or one positive–one negative. Plug in values for $$a$$ and $$b$$: * If $$a = -2$$ and $$b = -4$$, then $$\frac{1}{a + b} = -\frac{1}{6}$$, which is _not_ greater than zero. * If $$a = 3$$ and $$b = 2$$, then $$\frac{1}{a + b} = \frac{1}{5}$$, which _is_ greater than zero. No definite answer, so **Stat.(1) → Maybe → IS → BCE**.
Correct. [[snippet]] According to Stat. (1), $$a \gt b$$. However, that doesn't tell you whether $$a$$ and $$b$$ are positive or negative, or one positive–one negative. Plug in values for $$a$$ and $$b$$: * If $$a = -2$$ and $$b = -4$$, then $$\frac{1}{a + b} = -\frac{1}{6}$$, which is _not_ greater than zero. * If $$a = 3$$ and $$b = 2$$, then $$\frac{1}{a + b} = \frac{1}{5}$$, which _is_ greater than zero. No definite answer, so **Stat.(1) → Maybe → IS → BCE**. According to Stat. (2), $$\frac{1}{a} \gt 0$$. From here, it follows that $$\frac{1}{a}$$ is a positive fraction, so $$a$$ must be positive as well (otherwise, $$\frac{1}{a}$$ would be negative). However, that doesn't tell you anything about $$b$$. Again, plug in values for $$a$$ and $$b$$: * If $$a = 3$$ and $$b = 2$$, then $$\frac{1}{a + b} = \frac{1}{5}$$, which _is_ greater than zero. * If $$a = 3$$ and $$b = -7$$, then $$\frac{1}{a + b} = -\frac{1}{4}$$, which is _not_ greater than zero. No definite answer, so **Stat.(2) → Maybe → IS → CE**. According to Stat. (1+2), $$a \gt b$$ and $$\frac{1}{a} \gt 0$$. That still allows for $$b$$ to be either positive or negative. The same plug-ins we used for Stat. (2) apply here as well, so this is insufficient as well. **Stat.(1+2) → Maybe → IS → E**.
Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient to answer the question asked.
Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient to answer the question asked.
BOTH Statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.