Question Number 121783 by Ar Brandon last updated on 11/Nov/20
$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{integer}\:\mathrm{values}\:\mathrm{of}\:\mathrm{x}\:\mathrm{satisfying}\: \\ $$$$\mathrm{the}\:\mathrm{inequality}\:\mathrm{2x}+\mathrm{1}<\mathrm{2log}_{\mathrm{2}} \left(\mathrm{x}+\mathrm{3}\right)\:\mathrm{is}\:\_\_\_. \\ $$
Answered by TANMAY PANACEA last updated on 11/Nov/20
$$ \\ $$$${x}=\mathrm{1}\:\:{LHS}=\mathrm{3}\:\:{RHS}=\mathrm{4}\:\:{RHS}>{LHS} \\ $$$${x}=−\mathrm{1}\:{LHS}\:=−\mathrm{1}\:\:{RHS}=\mathrm{2}\:\:{RHS}>{LHS} \\ $$$${x}=\mathrm{5}\:{LHS}=\mathrm{11}\:\:{RHS}=\mathrm{6}\:\:{LHS}>{RHS} \\ $$$${x}=\mathrm{13}\:\:{L}=\mathrm{27}\:\:{R}=\mathrm{8}\:\:{L}>{R} \\ $$$${x}=\mathrm{29}\:\:{L}=\mathrm{59}\:\:{R}=\mathrm{10}\:\:{L}>{R} \\ $$$$\boldsymbol{{so}}\:\:\:\:\boldsymbol{{x}}=\mathrm{1}\:\boldsymbol{{and}}\:−\mathrm{1}\:\boldsymbol{{satisfy}}\:\boldsymbol{{the}}\:\boldsymbol{{eqn}} \\ $$
Commented by Ar Brandon last updated on 11/Nov/20
Thanks Sir.
But there are 4 integers according to the answer guide.
Commented by Dwaipayan Shikari last updated on 11/Nov/20
$$\mathrm{0}\:{and}\:−\mathrm{2}\:\left({Another}\:{two}\right) \\ $$