Question

Your friend says that the absolute value equation |3x+8|−9=−5 has no solution because the constant on the right side of the equation is negative. Is your friend correct? Explain.

Answer 1

$Solution, solve\:for\:x,\:\left|3x+8\right|-9=-5\quad :\quad x=-4\quad \mathrm{or}\quad \:x=-\frac{4}{3}$

$Steps:$

$\left|3x+8\right|-9=-5$

$\mathrm{Add\:}9\mathrm{\:to\:both\:sides},\\\left|3x+8\right|-9+9=-5+9$

$\mathrm{Simplify},\\\left|3x+8\right|=4$

$|f\left(x\right)|=a\quad \Rightarrow \:f\left(x\right)=-a\quad \mathrm{or}\quad \:f\left(x\right)=a,\\3x+8=-4\quad \quad \mathrm{or}\quad \:\quad \:3x+8=4$

$3x+8=-4,\\\mathrm{Subtract\:}8\mathrm{\:from\:both\:sides},\\3x+8-8=-4-8,\\\mathrm{Simplify},\\3x=-12,\\\mathrm{Divide\:both\:sides\:by\:}3,\\\frac{3x}{3}=\frac{-12}{3},\\\mathrm{Simplify},\\x=-4$

$3x+8=4,\\\mathrm{Subtract\:}8\mathrm{\:from\:both\:sides},\\3x+8-8=4-8,\\\mathrm{Simplify},\\3x=-4,\\\mathrm{Divide\:both\:sides\:by\:}3,\\\frac{3x}{3}=\frac{-4}{3},\\\mathrm{Simplify},\\x=-\frac{4}{3}$

$Combine\:the\:ranges,\\x=-4\quad \mathrm{or}\quad \:x=-\frac{4}{3}$

$\mathrm{The\:Correct\:Answer\:is\:No\:They\:Are\:Not\:Correct}$

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

$\mathrm{-Austint1414}$