How do you solve $(3x+5)/(6-2x)<=0$ ?

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Answer 1 · 2017-01-19T06:41:50

The answer is $x in ] -oo,-5/3 ] uu ] 3, +oo [$

Let $f(x)=(3x+5)/(6-2x)$

The domain of $f(x)$ is $D_f(x)=RR-{3}$

Now, we can make the sign chart

$color(white)(aaaa)$ $x$ $color(white)(aaaa)$ $-oo$ $color(white)(aaaa)$ $-5/3$ $color(white)(aaaaaaa)$ $3$ $color(white)(aaaaaaaaaaa)$ $+oo$

$color(white)(aaaa)$ $3x+5$ $color(white)(aaaa)$ $-$ $color(white)(aaaaaa)$ $+$ $color(white)(aa)$ $∥$ $color(white)(aaaa)$ $+$

$color(white)(aaaa)$ $6-2x$ $color(white)(aaaa)$ $+$ $color(white)(aaaaaa)$ $+$ $color(white)(aa)$ $∥$ $color(white)(aaaa)$ $-$

$color(white)(aaaa)$ $f(x)$ $color(white)(aaaaaa)$ $-$ $color(white)(aaaaaa)$ $+$ $color(white)(aa)$ $∥$ $color(white)(aaaa)$ $-$

Therefore,

$f(x)<=0$ , when $x in ] -oo,-5/3 ] uu ] 3, +oo [$

How do you solve (3x+5)/(6-2x)<=0(3x+5)/(6-2x)<=0?