Question Number 146636 by mathdanisur last updated on 14/Jul/21
$$\mathrm{4}\:+\:\frac{\mathrm{12}}{\mathrm{8}\:+\:\frac{\mathrm{7}}{\mathrm{3}\:+\:\frac{\mathrm{4}}{\boldsymbol{{x}}\:+\:\mathrm{1}}}}\:=\:\mathrm{8}\:\:\Rightarrow\:\:\boldsymbol{{x}}=? \\ $$
Commented by hknkrc46 last updated on 14/Jul/21
$$\bigstar\:\frac{\mathrm{12}}{\mathrm{8}\:+\:\frac{\mathrm{7}}{\mathrm{3}\:+\:\frac{\mathrm{4}}{\boldsymbol{{x}}\:+\:\mathrm{1}}}}\:=\:\mathrm{4}\: \\ $$$$\Rightarrow\:\mathrm{8}\:+\:\frac{\mathrm{7}}{\mathrm{3}\:+\:\frac{\mathrm{4}}{\boldsymbol{{x}}\:+\:\mathrm{1}}}\:=\:\mathrm{3} \\ $$$$\Rightarrow\:\frac{\mathrm{7}}{\mathrm{3}\:+\:\frac{\mathrm{4}}{\boldsymbol{{x}}\:+\:\mathrm{1}}}\:=\:−\mathrm{5} \\ $$$$\Rightarrow\:\mathrm{3}\:+\:\frac{\mathrm{4}}{\boldsymbol{{x}}\:+\:\mathrm{1}}\:=\:−\frac{\mathrm{7}}{\mathrm{5}} \\ $$$$\Rightarrow\:\frac{\mathrm{4}}{\boldsymbol{{x}}\:+\:\mathrm{1}}\:=\:−\frac{\mathrm{7}}{\mathrm{5}}−\mathrm{3}\:\Rightarrow\:\frac{\mathrm{4}}{\boldsymbol{{x}}\:+\:\mathrm{1}}\:=\:−\frac{\mathrm{22}}{\mathrm{5}} \\ $$$$\Rightarrow\:\mathrm{20}\:=\:−\mathrm{22}\boldsymbol{{x}}−\mathrm{22}\:\Rightarrow\:\mathrm{42}\:=\:−\mathrm{22}\boldsymbol{{x}} \\ $$$$\Rightarrow\:\boldsymbol{{x}}\:=\:−\frac{\mathrm{42}}{\mathrm{22}}\:=\:−\frac{\mathrm{21}}{\mathrm{11}} \\ $$
Commented by mathdanisur last updated on 14/Jul/21
$${cool}\:{thank}\:{you}\:{Ser} \\ $$
Answered by Olaf_Thorendsen last updated on 14/Jul/21
$$\mathrm{X}\:=\:\mathrm{4}+\frac{\mathrm{12}}{\mathrm{8}+\frac{\mathrm{7}}{\mathrm{3}+\frac{\mathrm{4}}{{x}+\mathrm{1}}}} \\ $$$$\mathrm{X}\:=\:\mathrm{4}+\frac{\mathrm{12}}{\mathrm{8}+\frac{\mathrm{7}}{\frac{\mathrm{3}{x}+\mathrm{7}}{{x}+\mathrm{1}}}} \\ $$$$\mathrm{X}\:=\:\mathrm{4}+\frac{\mathrm{12}}{\mathrm{8}+\frac{\mathrm{7}{x}+\mathrm{7}}{\mathrm{3}{x}+\mathrm{7}}} \\ $$$$\mathrm{X}\:=\:\mathrm{4}+\frac{\mathrm{12}}{\frac{\mathrm{31}{x}+\mathrm{63}}{\mathrm{3}{x}+\mathrm{7}}} \\ $$$$\mathrm{X}\:=\:\mathrm{4}+\frac{\mathrm{36}{x}+\mathrm{84}}{\mathrm{31}{x}+\mathrm{63}} \\ $$$$\mathrm{X}\:=\:\frac{\mathrm{160}{x}+\mathrm{336}}{\mathrm{31}{x}+\mathrm{63}} \\ $$$$\mathrm{X}\:=\:\mathrm{8}\:\Leftrightarrow\:\mathrm{160}{x}+\mathrm{336}\:=\:\mathrm{8}\left(\mathrm{31}{x}+\mathrm{63}\right) \\ $$$$\mathrm{160}{x}+\mathrm{336}\:=\:\mathrm{248}{x}+\mathrm{504} \\ $$$$\mathrm{88}{x}\:=\:−\mathrm{168} \\ $$$${x}\:=\:−\frac{\mathrm{168}}{\mathrm{88}}\:=\:−\frac{\mathrm{21}}{\mathrm{11}} \\ $$
Commented by mathdanisur last updated on 14/Jul/21
$${cool}\:{thank}\:{you}\:{Ser} \\ $$