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

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

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Answer 1 · 2016-12-16T17:03:54

$-5 < x <= 0$

Explanation:

$color(red)("There is a trap in this question in that there are excluded")$ $color(red)("values for "x)$

$color(blue)("Step 1")$
Solve disregarding the excluded values.

Multiply both sides by $(x+5)$

$2x<=0xx(x+5)$

$2x<=0$

$color(blue)("Condition 1: "x<=0)$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$color(blue)("Step 2")$
An equation or expression becomes 'undefined' if you have division by 0.

$=>(x+5) != 0$

$color(blue)("Condition 2: "x!=-5)$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$color(blue)("Step 3")$

Can we have $x< -5color(white)(.)?$

Suppose $x=-6$ then we have:

$(2(-6))/(-6+5) = (-12)/-1=+12$

This is not an excluded value but does not satisfy $(2x)/(x+5)<=0$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$color(blue)("Putting it all together using proper notation")$

$-5 < x <= 0$

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

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