Відповіді на вправи: 475

1) 6x ≤ 8x² – 2;

6x – 8x² + 2 ≤ 0 | • (–1);

8x² – 6x – 2 ≥ 0;

8x² – 6x – 2 = 0;

D = (–6)² – 4 • 8 • (–2) = 100 =

= 10²;

x₁ = $\frac{6-10}{16}$ = $\frac{-4}{16}$ = –$\frac{1}{4}$ = –0,25;

x₂ = $\frac{6+10}{16}$ = $\frac{16}{16}$ = 1;

x ∈ (–∞; –0,25] ∪ [1; +∞).

2) 3x² + 10 ≤ –x² + 22x;

3x² + 10 + x² – 22x ≤ 0;

4x² – 22x + 10 ≤ 0|: 2;

2x² – 11x + 5 ≤ 0;

2x² – 11x + 5 = 0;

D = (–11)² – 4 • 2 • 5 = 81 = 9²;

x₁ = $\frac{11+9}{4}$ = $\frac{20}{4}$ = 5;

x₂ = $\frac{11-9}{4}$ = $\frac{2}{4}$ = 0,5;

x ∈ [0,5; 5].

4) 3x – 5x² ≥ 3x² + 2;

–5x² – 3x² + 3x – 2 ≥ 0;

–8x² + 3x – 2 ≥ 0 |• (–1);

8x² – 3x + 2 ≤ 0;

8x² – 3x + 2 = 0;

D = (–3)² – 4 • 8 • 2 < 0;

∅

5) 9x² < –2x² – 12x – 1;

9x² + 2x² + 12x + 1 < 0;

11x² + 12x + 1 < 0;

D = 12² – 4 • 11 • 1 = 100 = 10²;

x₁ = $\frac{-12-10}{22}$ = $\frac{-22}{22}$ = –1;

x₂ = $\frac{-12+10}{22}$ = $\frac{-2}{22}$ = –$\frac{1}{11}$;

x ∈ (–1; –$\frac{1}{11}$).