$\text{a)}$  

${\left(7-x\right)}^2>{\left(x-6\right)}^2$  

$49-14x+x^2>x^2-12x+36\mid -x^2$  

$49-14x>-12x+36\mid +14x$  

$49>2x+36\mid -36$  

$13>2x\mid :2$  

$\frac{13}{2}>x$  

$x\in \left(-\infty ,\frac{13}{2}\right)$  

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$\text{b)}$  

$\left(8x+3\right)\left(\frac{1}{2}x-1\right)<{\left(2x-5\right)}^2-1$  

$4x^2-8x+\frac{3}{2}x-3<4x^2-20x+25-1$  

$4x^2-6\frac{1}{2}x-3<4x^2-20x+24\mid -4x^2$  

$-6\frac{1}{2}x-3<-20x+24\mid \cdot 2$  

$-13x-6<-40x+48\mid +40x$  

$27x-6<48\mid +6$  

$27x<54\mid :2$  

$x<2$  

$x\in \left(-\infty ,2\right)$  

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$\text{c)}$  

$24x+\left(6x-\sqrt{3}\right)\left(6x+\sqrt{3}\right)<{\left(6x+2\right)}^2$  

$24x+{\left(6x\right)}^2-({\sqrt{3}}^2<36x^2+24x+4$  

$24x+36x^2-3<36x^2+24x+4\mid -36x^2$  

$24x-3<24x+4\mid -24x$  

$-3<4$  

Nierówność prawdziwa.

$x\in \mathbb{R}$  

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$\text{d)}$  

$\frac{{\left(5x-3\right)}^2}{12}-\frac{2x-3}{3}\le \frac{{\left(5x+1\right)}^2}{12}-\frac{17x+11}{6}$  

$\frac{25x^2-30x+9}{12}-\frac{2x-3}{3}\le \frac{25x^2+10x+1}{12}-\frac{17x+11}{6}\mid \cdot 12$  

$25x^2-30x+9-4\left(2x-3\right)\le 25x^2+10x+1-2\left(17x+11\right)$  

$25x^2-30x+9-8x+12\le 25x^2+10x+1-34x-22\mid -25x^2$  

$-38x+21\le -24x-21\mid +38x$  

$21\le 14x-21\mid +21$  

$42\le 14x\mid :14$  

$3\le x$  

$x\in ⟨3,+\infty )$  

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$\text{e)}$  

${\left(x-2\right)}^3<x{\left(x-3\right)}^2$  

$x^3-6x^2+12x-8<x\left(x^2-6x+9\right)$  

$x^3-6x^2+12x-8<x^3-6x^2+9x\mid -x^3$  

$-6x^2+12x-8<-6x^2+9x\mid +6x^2$  

$12x-8<9x\mid -9x$  

$3x-8<0\mid +8$  

$3x<8\mid :3$  

$x<\frac{8}{3}$  

$x\in \left(-\infty ,\frac{13}{2}\right)$  

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$\text{f)}$  

$27x^2\left(x+1\right)+1>{\left(3x+1\right)}^3$  

$27x^3+27x^2+1>27x^3+27x^2+9x+1\mid -27x^3$  

$27x^2+1>27x^2+9x+1\mid -27x^2$  

$1>9x+1\mid -1$  

$0>9x\mid :9$  

$0>x$  

$x\in \left(-\infty ,0\right)$