How do solve $(3x+1)/(x+4)<=1$ and write the answer as a inequality and interval notation?

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How do solve $(3x+1)/(x+4)<=1$ and write the answer as a inequality and interval notation?

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Answer 1 · 2018-03-10T07:07:54

The solution is $x in (-4,3/2]$ or $-4 < x <= 3/2$

Explanation:

Let's rewrite and simplify the inequality

$(3x+1)/(x+4)<=1$

$(3x+1)/(x+4)-1<=0$

$((3x+1)-(x+4))/(x+4)<=0$

$((2x-3))/(x+4)<=0$

Let $f(x)=((2x-3))/(x+4)$

Now, we can build the sign chart

$color(white)(aaaa)$ $x$ $color(white)(aaaa)$ $-oo$ $color(white)(aaaaa)$ $-4$ $color(white)(aaaaaaa)$ $3/2$ $color(white)(aaaa)$ $+oo$

$color(white)(aaaa)$ $x+4$ $color(white)(aaaaaa)$ $-$ $color(white)(aa)$ $0$ $color(white)(aaaa)$ $+$ $color(white)(aaaa)$ $+$

$color(white)(aaaa)$ $2x-3$ $color(white)(aaaaa)$ $-$ $color(white)(aa)$ #color(white)(aaaaa)-# $color(white)(aa)$ $0$ $color(white)(aa)$ $+$

$color(white)(aaaa)$ $f(x)$ $color(white)(aaaaaaa)$ $+$ $color(white)(aa)$ $||$ $color(white)(aaaa)$ $-$ $color(white)(aa)$ $0$ $color(white)(aa)$ $+$

Therefore,

$f(x)<=0$ when $x in (-4,3/2]$ or $-4 < x <= 3/2$

graph{(3x+1)/(x+4)-1 [-27.09, 18.51, -12.22, 10.59]}

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How do solve $(3x+1)/(x+4)<=1$ and write the answer as a inequality and interval notation?

1 Answer

Explanation:

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How do solve $(3x+1)/(x+4)<=1$ and write the answer as a inequality and interval notation?