[tex]a)\\4(x-3)^2-(2x-5)^2\geq2\\4(x^2-6x+9)-(4x^2-20x+25)\geq2\\4x^2-24x+36-4x^2+20x-25\geq2\\4x^2-4x^2-24x+20x+36-25-2\geq0\\-4x+9\geq0\\-4x\geq-9 /:(-4)\\x\leq\frac94\\\\b)\\9(\frac23x-1)^2>(1-2x)^2-8x\\9(\frac49x^2-\frac43x+1)>1-4x+4x^2-8x\\4x^2-12x+9>1-12x+4x^2\\4x^2-4x^2-12x+12x+9-1>0\\\\8>0\\\text{Rownanie prawdziwe}\\x > 0 \text{dla }x \in R[/tex]
[tex]c)\\2(x+2)^2-(\sqrt2x-2)^2\geq0\\2(x^2+4x+4)-(2x^2-4\sqrt2x+4)\geq0\\2x^2+8x+8-2x^2+4\sqrt2x-4\geq0\\8x+4\sqrt2x+4\geq0\\x(8+4\sqrt2)+4\geq0\\x\g(8+4\sqrt2)\geq-4 /:(8+4\sqrt2)\\x\geq-\frac4{8+4\sqrt2}\\x\geq-\frac4{4(2+\sqrt2)}\\x\geq-\frac1{2+\sqrt2}*\frac{2-\sqrt2}{2-\sqrt2}\\x\geq-\frac{2-\sqrt2}{4-2}\\x\geq-\frac{2-\sqrt2}2[/tex]
[tex]d)\\-9(2-x)^2-(1-3x)(3x+1)\leq11\\-9(4-4x+x^2)-(3x+1-9x^2-3x)\leq11\\-36+36x-9x^2-(1-9x^2)\leq11\\-36+36x-9x^2-1+9x^2\leq11\\36x\leq11+36+1\\36x\leq48 /:36\\x\leq\frac{48}{36}\\x\leq\frac{4}3\\\\e)\\(\frac14x+2)^2+\frac14(1-\frac12x)(1+\frac12x)\geq0\\\frac1{16}x^2+x+4+\frac14(1-\frac14x^2)\geq0\\\frac1{16}x^2+x+4+\frac14-\frac1{16}x^2\geq0\\x\geq0-4-\frac14\\x\geq-4\frac14\\[/tex]
[tex]\\f)\\(\frac{\sqrt2}2x+1)(\frac{\sqrt2}2x-1)<\frac{(x-1)^2}2\\(\frac{\sqrt2}2x)^2-1<\frac{x^2-2x+1}2\\\frac24x^2-1<\frac{x^2-2x+1}2\\\frac12x^2-1<\frac{x^2-2x+1}2 /*2\\2(\frac12x^2-1)<x^2-2x+1\\x^2-2<x^2-2x+1\\x^2-x^2+2x<1+2\\2x<3 /:2\\x<\frac32[/tex]
