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Question Number 146636 by mathdanisur last updated on 14/Jul/21

4 + ((12)/(8 + (7/(3 + (4/(x + 1)))))) = 8 ⇒ x=?

$$\mathrm{4}\:+\:\frac{\mathrm{12}}{\mathrm{8}\:+\:\frac{\mathrm{7}}{\mathrm{3}\:+\:\frac{\mathrm{4}}{\boldsymbol{{x}}\:+\:\mathrm{1}}}}\:=\:\mathrm{8}\:\:\Rightarrow\:\:\boldsymbol{{x}}=? \\ $$

Commented by hknkrc46 last updated on 14/Jul/21

★ ((12)/(8 + (7/(3 + (4/(x + 1)))))) = 4 ⇒ 8 + (7/(3 + (4/(x + 1)))) = 3 ⇒ (7/(3 + (4/(x + 1)))) = −5 ⇒ 3 + (4/(x + 1)) = −(7/5) ⇒ (4/(x + 1)) = −(7/5)−3 ⇒ (4/(x + 1)) = −((22)/5) ⇒ 20 = −22x−22 ⇒ 42 = −22x ⇒ x = −((42)/(22)) = −((21)/(11))

$$\bigstar\:\frac{\mathrm{12}}{\mathrm{8}\:+\:\frac{\mathrm{7}}{\mathrm{3}\:+\:\frac{\mathrm{4}}{\boldsymbol{{x}}\:+\:\mathrm{1}}}}\:=\:\mathrm{4}\: \\ $$$$\Rightarrow\:\mathrm{8}\:+\:\frac{\mathrm{7}}{\mathrm{3}\:+\:\frac{\mathrm{4}}{\boldsymbol{{x}}\:+\:\mathrm{1}}}\:=\:\mathrm{3} \\ $$$$\Rightarrow\:\frac{\mathrm{7}}{\mathrm{3}\:+\:\frac{\mathrm{4}}{\boldsymbol{{x}}\:+\:\mathrm{1}}}\:=\:−\mathrm{5} \\ $$$$\Rightarrow\:\mathrm{3}\:+\:\frac{\mathrm{4}}{\boldsymbol{{x}}\:+\:\mathrm{1}}\:=\:−\frac{\mathrm{7}}{\mathrm{5}} \\ $$$$\Rightarrow\:\frac{\mathrm{4}}{\boldsymbol{{x}}\:+\:\mathrm{1}}\:=\:−\frac{\mathrm{7}}{\mathrm{5}}−\mathrm{3}\:\Rightarrow\:\frac{\mathrm{4}}{\boldsymbol{{x}}\:+\:\mathrm{1}}\:=\:−\frac{\mathrm{22}}{\mathrm{5}} \\ $$$$\Rightarrow\:\mathrm{20}\:=\:−\mathrm{22}\boldsymbol{{x}}−\mathrm{22}\:\Rightarrow\:\mathrm{42}\:=\:−\mathrm{22}\boldsymbol{{x}} \\ $$$$\Rightarrow\:\boldsymbol{{x}}\:=\:−\frac{\mathrm{42}}{\mathrm{22}}\:=\:−\frac{\mathrm{21}}{\mathrm{11}} \\ $$

Commented by mathdanisur last updated on 14/Jul/21

cool thank you Ser

$${cool}\:{thank}\:{you}\:{Ser} \\ $$

Answered by Olaf_Thorendsen last updated on 14/Jul/21

X = 4+((12)/(8+(7/(3+(4/(x+1)))))) X = 4+((12)/(8+(7/((3x+7)/(x+1))))) X = 4+((12)/(8+((7x+7)/(3x+7)))) X = 4+((12)/((31x+63)/(3x+7))) X = 4+((36x+84)/(31x+63)) X = ((160x+336)/(31x+63)) X = 8 ⇔ 160x+336 = 8(31x+63) 160x+336 = 248x+504 88x = −168 x = −((168)/(88)) = −((21)/(11))

$$\mathrm{X}\:=\:\mathrm{4}+\frac{\mathrm{12}}{\mathrm{8}+\frac{\mathrm{7}}{\mathrm{3}+\frac{\mathrm{4}}{{x}+\mathrm{1}}}} \\ $$$$\mathrm{X}\:=\:\mathrm{4}+\frac{\mathrm{12}}{\mathrm{8}+\frac{\mathrm{7}}{\frac{\mathrm{3}{x}+\mathrm{7}}{{x}+\mathrm{1}}}} \\ $$$$\mathrm{X}\:=\:\mathrm{4}+\frac{\mathrm{12}}{\mathrm{8}+\frac{\mathrm{7}{x}+\mathrm{7}}{\mathrm{3}{x}+\mathrm{7}}} \\ $$$$\mathrm{X}\:=\:\mathrm{4}+\frac{\mathrm{12}}{\frac{\mathrm{31}{x}+\mathrm{63}}{\mathrm{3}{x}+\mathrm{7}}} \\ $$$$\mathrm{X}\:=\:\mathrm{4}+\frac{\mathrm{36}{x}+\mathrm{84}}{\mathrm{31}{x}+\mathrm{63}} \\ $$$$\mathrm{X}\:=\:\frac{\mathrm{160}{x}+\mathrm{336}}{\mathrm{31}{x}+\mathrm{63}} \\ $$$$\mathrm{X}\:=\:\mathrm{8}\:\Leftrightarrow\:\mathrm{160}{x}+\mathrm{336}\:=\:\mathrm{8}\left(\mathrm{31}{x}+\mathrm{63}\right) \\ $$$$\mathrm{160}{x}+\mathrm{336}\:=\:\mathrm{248}{x}+\mathrm{504} \\ $$$$\mathrm{88}{x}\:=\:−\mathrm{168} \\ $$$${x}\:=\:−\frac{\mathrm{168}}{\mathrm{88}}\:=\:−\frac{\mathrm{21}}{\mathrm{11}} \\ $$

Commented by mathdanisur last updated on 14/Jul/21

cool thank you Ser

$${cool}\:{thank}\:{you}\:{Ser} \\ $$

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