Question

please help Solve |2b−9|=|b−6|

Answer 1

$Solution, solve\:for\:b,\:\left|2b-9\right|=\left|b-6\right|\quad :\quad b=3\quad \mathrm{or}\quad \:b=5$

$Steps:$

$\mathrm{Test\:each\:absolute\:for\:its\:positive\:and\:negative\:ranges},\\2b-9\ge \:0\:\mathrm{for}\:b\ge \frac{9}{2},\:\quad \mathrm{therefore\:for}\:b\ge \frac{9}{2}\quad \left|2b-9\right|=2b-9,\\2b-9<0\:\mathrm{for}\:b<\frac{9}{2},\:\quad \mathrm{therefore\:for}\:b<\frac{9}{2}\quad \left|2b-9\right|=-\left(2b-9\right),\\b-6\ge \:0\:\mathrm{for}\:b\ge \:6,\:\quad \mathrm{therefore\:for}\:b\ge \:6\quad \left|b-6\right|=b-6,$$b-6<0\:\mathrm{for}\:b<6,\:\quad \mathrm{therefore\:for}\:b<6\quad \left|b-6\right|=-\left(b-6\right)$

$\mathrm{Evaluate\:the\:expression\:in\:the\:following\:ranges:}, b<\frac{9}{2},\:\frac{9}{2}\le \:b<6,\:b\ge \:6$

$\mathrm{For:}\:b<\frac{9}{2},\\\mathrm{Replace:}\:\left|2b-9\right|\:\mathrm{with}\:-\left(2b-9\right),\\\mathrm{Replace:}\:\left|b-6\right|\:\mathrm{with}\:-\left(b-6\right),\\-\left(2b-9\right)=\left(-\left(b-6\right)\right)\quad :\quad b=3$

$\mathrm{For:}\:\frac{9}{2}\le \:b<6,\\\mathrm{Replace:}\:\left|2b-9\right|\:\mathrm{with}\:\left(2b-9\right),\\\mathrm{Replace:}\:\left|b-6\right|\:\mathrm{with}\:-\left(b-6\right),\\2b-9=\left(-\left(b-6\right)\right)\quad :\quad b=5$

$\mathrm{For:}\:b\ge \:6,\\\mathrm{Replace:}\:\left|2b-9\right|\:\mathrm{with}\:\left(2b-9\right),\\\mathrm{Replace:}\:\left|b-6\right|\:\mathrm{with}\:\left(b-6\right),\\2b-9=\left(b-6\right)\quad :\quad b=3$

$\mathrm{Combine\:the\:ranges},\\\left(b<\frac{9}{2}\quad \mathrm{and}\quad \:b=3\right)\quad \mathrm{or}\quad \left(\frac{9}{2}\le \:b<6\quad \mathrm{and}\quad \:b=5\right)\quad \mathrm{or}\quad \left(b\ge \:6\quad \mathrm{and}\quad \:b=3\right),\\

[tex]\mathrm{The\:Correct\:Answer\:is\:b=3\quad \mathrm{or}\quad \:b=5}$

$\mathrm{Hope\:This\:Helps!!!}$

$\mathrm{-Austint1414}$