Question

2) Resolver as inequações exponenciais:
a) 2˟⁺⁵ ≥ 64 b) 3˟ˉ¹ ≥ 27 c) 25˟ < 5)½ d) 8˟ˉ¹≤1/16˟ e) 2˟⁺⁷<32
me ajuda ☺

Answer 1

Resposta:

a) x ≥ 1

b) x ≥ 4

c) x < 1/4

d) x ≤ 3/7

e) x ≤ -2

Explicação passo-a-passo:

$a)\ 2^{x+5}\geq64\\.\ \ \ 2^{x+5}\geq2^{6}\longrightarrow x+5\geq6,\ logo:\ x\geq6-5\Longrightarrow \boxed{x\geq\bf{1}}$

$b)\ 3^{x-1}\geq27\\.\ \ \ 3^{x-1}\geq3^{3}\longrightarrow x-1\geq3,\ logo:\ x\geq3+1\Longrightarrow \boxed{x\geq\bf{4}}$

$c)\ 25^{x}<5^{\dfrac{1}{2}}\\.\ \ \ \left((5)^2\right)^{x}<5^{\dfrac{1}{2}}\longrightarrow 5^{2x}<5^{\dfrac{1}{2}},\ logo:\ 2x<\dfrac{1}{2}, ent\~{a}o:\ \boxed{x<\bf{\dfrac{1}{4}}}$

$d)\ 8^{x-1}\leq\dfrac{1}{16^{x}}\\\\.\ \ \left(2^3\right)^{x-1}\leq16^{-x}\\\\.\ \ \left(2^3\right)^{x-1}\leq\left(2^4\right)^{-x}\\\\.\ \ 2^{3x-3}\leq2^{-4x} \longrightarrow 3x-3\leq-4x,\ logo:\ 7x\leq\ 3 \Longrightarrow \boxed{x\leq\bf{\dfrac{3}{7}}}$

$e)\ 2^{x+7}\leq32\\.\ \ \ 2^{x+7}\leq2^{5}\longrightarrow x+7\leq5,\ logo:\ x\leq 5-7\Longrightarrow \boxed{x\leq\bf{-2}}$

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