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Question Number 121783 by Ar Brandon last updated on 11/Nov/20
The number of integer values of x satisfying   the inequality 2x+1<2log_2 (x+3) is ___.
$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{integer}\:\mathrm{values}\:\mathrm{of}\:\mathrm{x}\:\mathrm{satisfying}\: \\ $$$$\mathrm{the}\:\mathrm{inequality}\:\mathrm{2x}+\mathrm{1}<\mathrm{2log}_{\mathrm{2}} \left(\mathrm{x}+\mathrm{3}\right)\:\mathrm{is}\:\_\_\_. \\ $$
Answered by TANMAY PANACEA last updated on 11/Nov/20
  x=1  LHS=3  RHS=4  RHS>LHS  x=−1 LHS =−1  RHS=2  RHS>LHS  x=5 LHS=11  RHS=6  LHS>RHS  x=13  L=27  R=8  L>R  x=29  L=59  R=10  L>R  so    x=1 and −1 satisfy the eqn
$$ \\ $$$${x}=\mathrm{1}\:\:{LHS}=\mathrm{3}\:\:{RHS}=\mathrm{4}\:\:{RHS}>{LHS} \\ $$$${x}=−\mathrm{1}\:{LHS}\:=−\mathrm{1}\:\:{RHS}=\mathrm{2}\:\:{RHS}>{LHS} \\ $$$${x}=\mathrm{5}\:{LHS}=\mathrm{11}\:\:{RHS}=\mathrm{6}\:\:{LHS}>{RHS} \\ $$$${x}=\mathrm{13}\:\:{L}=\mathrm{27}\:\:{R}=\mathrm{8}\:\:{L}>{R} \\ $$$${x}=\mathrm{29}\:\:{L}=\mathrm{59}\:\:{R}=\mathrm{10}\:\:{L}>{R} \\ $$$$\boldsymbol{{so}}\:\:\:\:\boldsymbol{{x}}=\mathrm{1}\:\boldsymbol{{and}}\:−\mathrm{1}\:\boldsymbol{{satisfy}}\:\boldsymbol{{the}}\:\boldsymbol{{eqn}} \\ $$
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
0 and −2 (Another two)
$$\mathrm{0}\:{and}\:−\mathrm{2}\:\left({Another}\:{two}\right) \\ $$

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