How do you solve $\frac{a - 1}{a} > 0$ $(a-1)/a>0$ using a sign chart?

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How do you solve $\frac{a - 1}{a} > 0$ $(a-1)/a>0$ using a sign chart?

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Answer 1 · 2017-10-14T06:25:20

The solution is $a \in (- \infty, 0) \cup (1, + \infty)$ $a in (-oo,0) uu (1,+oo)$

Explanation:

Let $f (a) = \frac{a - 1}{a}$ $f(a)=(a-1)/a$

The sign chart is

$a a a a$ $color(white)(aaaa)$ $a$ $a$ $a a a a$ $color(white)(aaaa)$ $- \infty$ $-oo$ $a a a a a a a a$ $color(white)(aaaaaaaa)$ $0$ $0$ $a a a a a a a a$ $color(white)(aaaaaaaa)$ $1$ $1$ $a a a a$ $color(white)(aaaa)$ $+ \infty$ $+oo$

$a a a a$ $color(white)(aaaa)$ $a$ $a$ $a a a a a a a a a$ $color(white)(aaaaaaaaa)$ $-$ $-$ $a a a a$ $color(white)(aaaa)$ $∣ ∣$ $||$ $a a a a$ $color(white)(aaaa)$ $+$ $+$ $a a a a$ $color(white)(aaaa)$ $+$ $+$

$a a a a$ $color(white)(aaaa)$ $a - 1$ $a-1$ $a a a a a a$ $color(white)(aaaaaa)$ $-$ $-$ $a a a a$ $color(white)(aaaa)$ #color(white)(aaaaa)-# $a a a a$ $color(white)(aaaa)$ $+$ $+$

$a a a a$ $color(white)(aaaa)$ $f (a)$ $f(a)$ $a a a a a a a$ $color(white)(aaaaaaa)$ $+$ $+$ $a a a a$ $color(white)(aaaa)$ $∣ ∣$ $||$ $a a a a$ $color(white)(aaaa)$ $-$ $-$ $a a a a$ $color(white)(aaaa)$ $+$ $+$

Therefore,

$f (a) > 0$ $f(a)>0$ , when $a \in (- \infty, 0) \cup (1, + \infty)$ $a in (-oo,0) uu (1,+oo)$

graph{(x-1)/x [-16.02, 16.01, -8.01, 8.01]}

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