How do you solve $1/4<7/(7-x)$ using a sign chart?

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Answer 1 · 2017-06-06T18:42:47

The solution is $x in (-21,7)$

We cannot do crossing over.

Let's rearrange the equation

$1/4<7/(7-x)$

$7/(7-x)-1/4>0$

$(28-(7-x))/(4(7-x))>0$

$(21+x)/(4(7-x))>0$

Let $f(x)=(21+x)/(4(7-x))$

Let's build the sign chart

$color(white)(aaaa)$ $x$ $color(white)(aaaa)$ $-oo$ $color(white)(aaaa)$ $-21$ $color(white)(aaaaaaa)$ $7$ $color(white)(aaaaaaa)$ $+oo$

$color(white)(aaaa)$ $21+x$ $color(white)(aaaaa)$ $-$ $color(white)(aaaa)$ $+$ $color(white)(aaaa)$ $||$ $color(white)(aaaa)$ $+$

$color(white)(aaaa)$ $7-x$ $color(white)(aaaaaa)$ $-$ $color(white)(aaaa)$ $+$ $color(white)(aaaa)$ $||$ $color(white)(aaaa)$ $-$

Therefore,

$f(x)>0$ when $x in (-21,7)$

How do you solve 1/4<7/(7-x)1/4<7/(7-x) using a sign chart?