How do you solve $3x-5 <2x-5 <5x-2$ ?

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Answer 1 · 2017-02-26T20:14:21

$-1 < x < 0$

Given: $" "3x-5<2x-5<5x-2" "$ Inequality(1)

Lets get rid of one of the $x$ terms

Subtract $2x$ from every part

$x-5<-5<3x-2$

Add $5$ to very part

$x<0<3x+3 " Inequality(2)"$
~~~~~~~~~~~~~~~~~~~~~~~~~~

$color(brown)("Observation(1): So "x" is negative and "3x+3>0)$

Investigate $3x+3>0$

Factor out the 3

$3(x+1)>0$

Divide both sides by 3. Note that $0/3=0$

$x+1>0$

$x > -1" "$ Inequality(3)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$color(brown)("Observation(2): Combining Inequality(3) with Inequality(2)")$

$-1 < x < 0$

How do you solve 3x-5 <2x-5 <5x-23x-5 <2x-5 <5x-2?