Відповіді на вправи: 479

1) $\frac{{2x}^2+1}{3}$ < $\frac{3x+5}{2}$ |• 6;

2(2x² + 1) < 3(3х + 5);

4x² + 2 < 9x + 15;

4x – 9x – 13 < 0;

4x² – 9x – 13 = 0;

D = (–9)² – 4 • 4 • (–13) = 289 = 17²;

x₁ = $\frac{9-17}{8}$ = $\frac{-8}{8}$ = –1;

x₂ = $\frac{9+17}{8}$ = $\frac{26}{8}$ = $\frac{13}{4}$= 3,25;

2) $\frac{x^2+3x}{6}$ $\ge$ $\frac{4x-2}{3}$ |• 6;

x² + 3x $\ge$ 2(4x – 2);

x² + 3х $\ge$ 8х – 4;

x² + 3х – 8х + 4 $\ge$ 0;

x² – 5x + 4 $\ge$ 0;

x² – 5x + 4 = 0;

x₁ = 1, x₂ = 4.

х $\in$ (–$\infty$; 1] $\cup$ [4; +$\infty$).

3) $\frac{{2x}^2+3х}{4}$ $\le$ $\frac{3x+1}{2}$ |• 4;

2x² + 3x $\le$ 2(3х + 1);

2x² + 3х $\le$ 6x + 2;

2x² + 3X – 6x – 2 $\le$ 0;

2x² – 3x – 2 = 0;

D = (–3)² – 4 • 2 • (–2) = 25 = 5²;

x₁ = $\frac{3-5}{4}$ = $\frac{-2}{4}$ = –$\frac{1}{2}$;

x₂ = $\frac{3+5}{4}$ = $\frac{8}{4}$ = 2.

[–$\frac{1}{2}$; 2].

4) $\frac{{5x}^2-х}{2}$ > х + 1 |• 2;

5x² – х > 2(x + 1);

5x² – x > 2x + 2;

5x² – x – 2x – 2 > 0;

5x² – 3x – 2 = 0;

D = (–3)² – 4 • 5 • (–2) = 49 = 7²;

x₁ = $\frac{3-7}{10}$ = $\frac{-4}{10}$ = –0,4;

x₂ = $\frac{3+7}{10}$ = $\frac{10}{10}$ = 1.

х $\in$ (–$\infty$; 0,4) $\cup$ (1; +$\infty$).