Question Number 164269 by Zaynal last updated on 15/Jan/22
![Solve the inequality; (x+3)[(x−1)^2 −x(x+(3/4))]+6+((7x^2 +29x)/4)>0 {Z.A}](Q164269.png)
$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{Solve}}\:\boldsymbol{{the}}\:\boldsymbol{{inequality}}; \\ $$ $$\:\:\left(\boldsymbol{{x}}+\mathrm{3}\right)\left[\left(\boldsymbol{{x}}−\mathrm{1}\right)^{\mathrm{2}} −\boldsymbol{{x}}\left(\boldsymbol{{x}}+\frac{\mathrm{3}}{\mathrm{4}}\right)\right]+\mathrm{6}+\frac{\mathrm{7}\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{29}\boldsymbol{{x}}}{\mathrm{4}}>\mathrm{0} \\ $$ $$\left\{\boldsymbol{{Z}}.\boldsymbol{{A}}\right\} \\ $$
Answered by manxsolar last updated on 15/Jan/22
![primer termino raiz x=−3 segundl termino 24+7x^2 +29x=(7x+8)(x+3)/4 factorizando (x+3)[x^2 −2x+1−x^2 −((3x)/4)+((7x)/4)+(8/4)]⟩0 (x+3)[−2x+1+x+2]⟩0 (x+3)[−x+3]⟩0 por −1 (x+3)(x−3)⟨0 puntoz criticos x=−3 x..=3 −−−−−−−(−3)−−−−−−−(+3)−−−−−−− + − + me piden f(x)⟨0 entonces x∈<−3;3> sean x1,x2,x3,........xn raices del polinomio=(x−x1)(x−x2((x−x3)..... xn..........x3.......x2.........x1......... f(x) + − + condiciones uso termino x positivo x1,x2,x3, ordenadoz eje carteziano](Q164275.png)
$${primer}\:{termino}\:{raiz}\:{x}=−\mathrm{3} \\ $$ $${segundl}\:{termino}\:\:\mathrm{24}+\mathrm{7}{x}^{\mathrm{2}} +\mathrm{29}{x}=\left(\mathrm{7}{x}+\mathrm{8}\right)\left({x}+\mathrm{3}\right)/\mathrm{4} \\ $$ $${factorizando} \\ $$ $$\:\:\:\left({x}+\mathrm{3}\right)\left[{x}^{\mathrm{2}} −\mathrm{2}{x}+\mathrm{1}−{x}^{\mathrm{2}} −\frac{\mathrm{3}{x}}{\mathrm{4}}+\frac{\mathrm{7}{x}}{\mathrm{4}}+\frac{\mathrm{8}}{\mathrm{4}}\right]\rangle\mathrm{0} \\ $$ $$\left({x}+\mathrm{3}\right)\left[−\mathrm{2}{x}+\mathrm{1}+{x}+\mathrm{2}\right]\rangle\mathrm{0} \\ $$ $$\left({x}+\mathrm{3}\right)\left[−{x}+\mathrm{3}\right]\rangle\mathrm{0} \\ $$ $${por}\:−\mathrm{1} \\ $$ $$\left({x}+\mathrm{3}\right)\left({x}−\mathrm{3}\right)\langle\mathrm{0} \\ $$ $${puntoz}\:{criticos} \\ $$ $${x}=−\mathrm{3} \\ $$ $${x}..=\mathrm{3} \\ $$ $$−−−−−−−\left(−\mathrm{3}\right)−−−−−−−\left(+\mathrm{3}\right)−−−−−−− \\ $$ $$\:\:\:\:\:\:\:\:\:\:\:\:+\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+ \\ $$ $${me}\:{piden}\:{f}\left({x}\right)\langle\mathrm{0} \\ $$ $${entonces}\:\:\:\:\:\:\:\:\:\:\:\:{x}\in<−\mathrm{3};\mathrm{3}> \\ $$ $${sean}\:{x}\mathrm{1},{x}\mathrm{2},{x}\mathrm{3},........{xn}\:\:\:\:{raices}\:{del}\:{polinomio}=\left({x}−{x}\mathrm{1}\right)\left({x}−{x}\mathrm{2}\left(\left({x}−{x}\mathrm{3}\right).....\right.\right. \\ $$ $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{xn}..........{x}\mathrm{3}.......{x}\mathrm{2}.........{x}\mathrm{1}......... \\ $$ $$\:\:\:\:\:\:\:{f}\left({x}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−\:\:\:\:\:\:\:\:\:\:\:\:\:\:+ \\ $$ $$ \\ $$ $$ \\ $$ $${condiciones}\:{uso} \\ $$ $${termino}\:{x}\:\:{positivo} \\ $$ $${x}\mathrm{1},{x}\mathrm{2},{x}\mathrm{3},\:\:\:\:\:{ordenadoz}\:{eje}\:{carteziano} \\ $$ $$ \\ $$ $$ \\ $$ $$ \\ $$ $$ \\ $$ $$ \\ $$ $$ \\ $$ $$ \\ $$ $$ \\ $$ $$ \\ $$ $$ \\ $$ $$ \\ $$ $$ \\ $$