Question

Resolva, em ℝ, as inequações. (cálculos)

b) (4x + 13)(3 − x)(2x − 1) ≤ 0

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Answer 1

$\boxed{\begin{array}{l}\boldsymbol{Resposta:}\\\sf S=\bigg\{x\in\mathbb{R}/-\dfrac{13}{4}\leqslant x\leqslant\dfrac{1}{2}~ou~x\geqslant3\bigg\}\\\boldsymbol{ Explicac_{\!\!,}\tilde ao~passo~a~passo:}\\\sf interpretemos~cada~parcela\\\sf~da~inequac_{\!\!,}\tilde ao-produto~como~func_{\!\!,}\tilde oes\\\sf Vamos~fazer~o~estudo~do~sinal\\\sf de~cada~uma,elaborar~um~quadro~de~sinais\\\sf e~por~fim~assinalar~o~intervalo~negativo\\\sf correspondente~a~soluc_{\!\!,}\tilde ao~da~inequac_{\!\!,}\tilde ao.\end{array}}$

$\large\boxed{\begin{array}{l}\sf (4x+13)(3-x)(2x-1)\leqslant0\\\sf f(x)=4x+13\\\underline{\rm ra\acute izes~de~f(x):}\\\sf 4x=-13\\\sf x=-\dfrac{13}{4}\\\sf f(x)>0~se~x>-\dfrac{13}{4}\\\\\sf f(x)<0~se~x<-\dfrac{13}{4}\\\sf g(x)=3-x\\\underline{\rm ra\acute izes~de~g(x):}\\\sf 3-x=0\\\sf x=3\\\sf g(x)>0~se~x<3~e~g(x)<0~se~x>3\\\sf h(x)=2x-1\\\underline{\rm ra\acute izes~de~h(x):}\\\sf 2x-1=0\\\sf 2x=1\\\sf x=\dfrac{1}{2}\\\sf h(x)>0~se~x>\dfrac{1}{2}\\\\\sf h(x)<0~se~x<\dfrac{1}{2}\end{array}}$

$\large\boxed{\begin{array}{l}\underline{\rm Observe~a~figura~que~eu~anexei:}\\\sf Assinalando~o~intervalo\\\sf negativo~temos~que\\\sf S=\bigg\{x\in\mathbb{R}/-\dfrac{13}{4}\leqslant x\leqslant\dfrac{1}{2}~ou~x\geqslant3\bigg\}\end{array}}$