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solve-x-3-3x-2-1-




Question Number 24663 by NECx last updated on 24/Nov/17
solve ∣x−3∣=∣3x+2∣−1
$$\boldsymbol{{solve}}\:\mid{x}−\mathrm{3}\mid=\mid\mathrm{3}{x}+\mathrm{2}\mid−\mathrm{1} \\ $$
Answered by ajfour last updated on 24/Nov/17
Case I:       x < −(2/3)  , then       3−x = −3x−2−1  or    2x =−6         ⇒  x=−3  Case II:       −(2/3) < x < 3       3−x = 3x+2−1  or     4x=2      ⇒  x = (1/2) .  Case III:      x > 3      x−3 = 3x+2−1    or    2x =−4     ⇒  x=−2   but this is not within considered  interval so gets rejected.  So finally    x= −3, (1/2) .
$${Case}\:{I}:\:\:\:\:\:\:\:{x}\:<\:−\frac{\mathrm{2}}{\mathrm{3}}\:\:,\:{then} \\ $$$$\:\:\:\:\:\mathrm{3}−{x}\:=\:−\mathrm{3}{x}−\mathrm{2}−\mathrm{1} \\ $$$${or}\:\:\:\:\mathrm{2}{x}\:=−\mathrm{6} \\ $$$$\:\:\:\:\:\:\:\Rightarrow\:\:\boldsymbol{{x}}=−\mathrm{3} \\ $$$${Case}\:{II}:\:\:\:\:\:\:\:−\frac{\mathrm{2}}{\mathrm{3}}\:<\:{x}\:<\:\mathrm{3} \\ $$$$\:\:\:\:\:\mathrm{3}−{x}\:=\:\mathrm{3}{x}+\mathrm{2}−\mathrm{1} \\ $$$${or}\:\:\:\:\:\mathrm{4}{x}=\mathrm{2} \\ $$$$\:\:\:\:\Rightarrow\:\:\boldsymbol{{x}}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\:. \\ $$$${Case}\:{III}:\:\:\:\:\:\:{x}\:>\:\mathrm{3} \\ $$$$\:\:\:\:{x}−\mathrm{3}\:=\:\mathrm{3}{x}+\mathrm{2}−\mathrm{1} \\ $$$$\:\:{or}\:\:\:\:\mathrm{2}{x}\:=−\mathrm{4}\:\:\:\:\:\Rightarrow\:\:{x}=−\mathrm{2}\: \\ $$$${but}\:{this}\:{is}\:{not}\:{within}\:{considered} \\ $$$${interval}\:{so}\:{gets}\:{rejected}. \\ $$$${So}\:{finally}\:\:\:\:\boldsymbol{{x}}=\:−\mathrm{3},\:\frac{\mathrm{1}}{\mathrm{2}}\:. \\ $$
Commented by NECx last updated on 25/Nov/17
thanks boss
$${thanks}\:{boss} \\ $$

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