Question

cine ma pote ajuta si pe mine cu un exercitiu la mate??
Sa se determine solutile intregi ale inecuatiei (x-1) la patrat+x-7<0

Answer 1

$(x-1)^2+x-7\ \textless \ 0 \\ \\ x^2-2x+1+x-7\ \textless \ 0 \\ \\ x^2-x-6\ \textless \ 0 \\ \\ (x^2-4)-x-2\ \textless \ 0 \\ \\ (x+2)(x-2)-(x+2)\ \textless \ 0 \\ \\ (x+2)(x-2-1)\ \textless \ 0 \\ \\ (x+2)(x-3)\ \textless \ 0.$

$Produsul~a~doua~numere~este~mai~mic~decat~0~atunci~cand~numerele \\ \\ au~semne~diferite~(unul~pozitiv~si~celalalt~negativ).$

$\underline{Cazul~I}:~x+2\ \textgreater \ 0~si~x-3\ \textless \ 0. \\ \\ x+2\ \textgreater \ 0 \Rightarrow x\ \textgreater \ -2. \\ \\ x-3\ \textless \ 0 \Rightarrow x\ \textless \ 3. \\ \\ x \in (-2;3). \\ \\ \underline{Cazul~II}:~x+2\ \textless \ 0~si~x-3\ \textgreater \ 0. \\ \\ x+2\ \textless \ 0 \Rightarrow x\ \textless \ -2. \\ \\ x-3\ \textgreater \ 0 \Rightarrow x\ \textgreater \ 3. \\ \\ x\ \textless \ -2~si~x\ \textgreater \ 3~este~imposibil \Rightarrow~acest~caz~nu~are~solutie.$

$\underline{Solutie}: x \in (-2;3).$

Answer 2

(x-1)²+x-7<0
x²-2x+1+x-7=0
x²-x-6=0
a=1
b=-1
c=-6
Δ=b²-4ac
Δ=1+24=25
Δ=25>0
x₁=(1-5)/2=-4/2=-2
x₂=(1+5)/2=6/2=3

x            -∞                   -2                         3                       +∞
x²-x-6     ++++++++++++0----------------------0++++++++++++++
S=(-2,3)