Question

Rezolvati in multimea nr. naturale urm inecuatii:
a) 4(x-1)-2(x+1) b) 2(x+1)²+3(x-2)≥3(x+2)(x-2)-(x²-8x)

Answer 1

$\it b) \ 2(x+1)^2+3(x-2)\geq3(x+2)(x-2)-(x^2-8x) \Rightarrow \\ \\ \Rightarrow 2(x^2+2x+1) +3x-6 \geq3(x^2-4) -x^2+8x \Rightarrow \\ \\ \Rightarrow 2x^2 +4x+2 +3x-6 \geq3x^2-12-x^2+8x \Rightarrow \\ \\ \Rightarrow 2x^2+7x-4\geq2x^2+8x-12 \Rightarrow -4+12\geq8x-7x \Rightarrow \\ \\ \Rightarrow 8\geq x \Rightarrow x\leq8 \Rightarrow x\in\{0,\ 1,\ 2,\ ...,\ 8\}$

Answer 2

a)

4(x-1)-2(x+1)

4x-4-2x-2

(4x-2x)+(-4-2)=2x-6

b)

2(x+1)²+3(x-2)≥3(x+2)(x-2)-(x²-8x)

2x²+4x+2+3x-6≥3x²-6x+6x-12-x²+8x

2x²+7x-4≥3x²-6x+6x-12-x²+8x

2x²+7x-4≥3x²-12-x²+8x

2x²+7x-4≥2x²-12+8x

7x-4≥-12+8x

7x-4≥8x-12

-4≥8x-12-7x

-4≥x-12

-4+12≥x

8≥x

x≤8