1) 5x-5+7≤1-3x-6
5x+2≤3x-5.
2)4a+32-7a+7<12
-3a+39<12.
3)4b-6-1.2≥6b-1
4b-7.2≥6b-1
4) 1.7-3+3m≤-m+1.9
-1.3+3m≤m+1.9
Точки пересечения параболы у=2-х² и прямой у=-х (биссектриса 2 и 4 координатных углов):
2-х²=-х
х²-х-2=0
По теореме Виета х₁=-1 , х₂=2
Область находится между параболой и прямой, причём на промежутке (-1,2) парабола лежит выше прямой. Площадь
S=(от -1 до 2) ∫ [ (2-х²) -(-х) ]dx=[2x-x³/3+x²/2] (подстановка от -1 до 2)=(2*2-2³/3+2²/2)-(-2+1/3+1/2)=(4-8/3+4/2) +2-1/3-1/2=6-9/3+3/2=6-3+3/2=3+1,5=4,5
Х это часть 5х-3х= 8 2х=8 х=4
3*4=12 км от А доБ
5*4=20 км отАдоВ
2а)
![\sin(x - \frac{\pi}{3} ) \geqslant \frac{ \sqrt{3} }{2} \\ \frac{ \pi}{3} + 2\pi \times n \leqslant x - \frac{\pi}{3} \leqslant \frac{2\pi}{3} + 2\pi \times n \\ \frac{ \pi}{3} +\frac{ \pi}{3} + 2\pi \times n \leqslant x - \frac{\pi}{3} + \frac{ \pi}{3}\leqslant \frac{2\pi}{3} +\frac{ \pi}{3} + 2\pi \times n \\ \frac{ 2\pi}{3} + 2\pi \times n \leqslant x\leqslant \pi+ 2\pi \times n](https://tex.z-dn.net/?f=+%5Csin%28x+-++%5Cfrac%7B%5Cpi%7D%7B3%7D+%29++%5Cgeqslant++%5Cfrac%7B+%5Csqrt%7B3%7D+%7D%7B2%7D++%5C%5C++%5Cfrac%7B+%5Cpi%7D%7B3%7D++%2B+2%5Cpi+%5Ctimes+n+%5Cleqslant+x+-++%5Cfrac%7B%5Cpi%7D%7B3%7D++%5Cleqslant++%5Cfrac%7B2%5Cpi%7D%7B3%7D++%2B+2%5Cpi+%5Ctimes+n+%5C%5C+++%5Cfrac%7B+%5Cpi%7D%7B3%7D+%2B%5Cfrac%7B+%5Cpi%7D%7B3%7D+++%2B+2%5Cpi+%5Ctimes+n+%5Cleqslant+x+-++%5Cfrac%7B%5Cpi%7D%7B3%7D+++%2B+%5Cfrac%7B+%5Cpi%7D%7B3%7D%5Cleqslant++%5Cfrac%7B2%5Cpi%7D%7B3%7D+++%2B%5Cfrac%7B+%5Cpi%7D%7B3%7D+%2B+2%5Cpi+%5Ctimes+n+%5C%5C+%5Cfrac%7B+2%5Cpi%7D%7B3%7D++%2B+2%5Cpi+%5Ctimes+n+%5Cleqslant+x%5Cleqslant++%5Cpi%2B+2%5Cpi+%5Ctimes+n)
nєZ.
xє[2π/3 + 2πn; π+2πn], nєZ.
2b)
![\cos(2x + \frac{\pi}{4} ) \leqslant - \frac{ \sqrt{2} }{2} \\ \frac{3\pi}{4} + 2\pi \times n \leqslant 2x + \frac{\pi}{4} \leqslant \frac{5\pi}{4} + 2\pi \times n \\ \frac{3\pi}{4} - \frac{\pi}{4} + 2\pi \times n \leqslant 2x + \frac{\pi}{4} - \frac{\pi}{4} \leqslant \frac{5\pi}{4} - \frac{\pi}{4} + 2\pi \times n \\ \frac{\pi}{2} + 2\pi \times n \leqslant 2x \leqslant \pi+ 2\pi \times n \: \: \: \: ( \div 2) \\ \frac{\pi}{4} + \pi \times n \leqslant x \leqslant \frac{\pi}{2} + \pi \times n \\](https://tex.z-dn.net/?f=+%5Ccos%282x+%2B++%5Cfrac%7B%5Cpi%7D%7B4%7D+%29++%5Cleqslant++-++%5Cfrac%7B+%5Csqrt%7B2%7D+%7D%7B2%7D++%5C%5C++%5Cfrac%7B3%5Cpi%7D%7B4%7D++%2B+2%5Cpi+%5Ctimes+n+%5Cleqslant++2x+%2B++%5Cfrac%7B%5Cpi%7D%7B4%7D++%5Cleqslant+%5Cfrac%7B5%5Cpi%7D%7B4%7D++%2B+2%5Cpi+%5Ctimes+n+%5C%5C+%5Cfrac%7B3%5Cpi%7D%7B4%7D++-++%5Cfrac%7B%5Cpi%7D%7B4%7D++%2B+2%5Cpi+%5Ctimes+n+%5Cleqslant++2x+%2B++%5Cfrac%7B%5Cpi%7D%7B4%7D+-++%5Cfrac%7B%5Cpi%7D%7B4%7D+++%5Cleqslant+%5Cfrac%7B5%5Cpi%7D%7B4%7D++-++%5Cfrac%7B%5Cpi%7D%7B4%7D++%2B+2%5Cpi+%5Ctimes+n+%5C%5C+%5Cfrac%7B%5Cpi%7D%7B2%7D++%2B+2%5Cpi+%5Ctimes+n+%5Cleqslant++2x++%5Cleqslant+%5Cpi%2B+2%5Cpi+%5Ctimes+n++%5C%3A++%5C%3A++%5C%3A++%5C%3A+%28+%5Cdiv+2%29+%5C%5C+%5Cfrac%7B%5Cpi%7D%7B4%7D++%2B+%5Cpi+%5Ctimes+n+%5Cleqslant++x+%5Cleqslant+%5Cfrac%7B%5Cpi%7D%7B2%7D++%2B+%5Cpi+%5Ctimes+n+%5C%5C+)
nєZ.
xє[π/4 +πn; π/2+πn], nєZ.
3.
![2 \sin^{2}(x) - \cos(x) > 2 \\ \sin^{2}(x) = 1 - \cos^{2}(x) \\ 2(1 - \cos^{2}(x)) - \cos(x) > 2 \\ 2 - 2\cos^{2}(x) - \cos(x) > 2 \\ 2 - 2\cos^{2}(x) - \cos(x) - 2 > 0 \\ - 2\cos^{2}(x) - \cos(x) > 0 \: \: \: (\times - 1) \\ 2\cos^{2}(x) + \cos(x) < 0 \\ \cos(x) (2 \cos(x) + 1) < 0 \\](https://tex.z-dn.net/?f=2+%5Csin%5E%7B2%7D%28x%29+-+%5Ccos%28x%29+%3E+2+%5C%5C+%5Csin%5E%7B2%7D%28x%29+%3D+1+-+%5Ccos%5E%7B2%7D%28x%29+%5C%5C+2%281+-+%5Ccos%5E%7B2%7D%28x%29%29+-++%5Ccos%28x%29++%3E+2+%5C%5C+2+-+2%5Ccos%5E%7B2%7D%28x%29+-+%5Ccos%28x%29++%3E+2++%5C%5C+2+-+2%5Ccos%5E%7B2%7D%28x%29+-+%5Ccos%28x%29+++-++2+%3E+0+%5C%5C+-+2%5Ccos%5E%7B2%7D%28x%29+-+%5Ccos%28x%29++%3E+0+++%5C%3A++%5C%3A++%5C%3A++%28%5Ctimes++-+1%29+%5C%5C+2%5Ccos%5E%7B2%7D%28x%29++%2B+%5Ccos%28x%29+++%3C++0+++%5C%5C++%5Ccos%28x%29++%282+%5Ccos%28x%29+++%2B+1%29+%3C+0+%5C%5C+)
найдем нули функции
соs(x)=0 при х1=π/2 +2πn, nєZ,
x2=3π/2 +2πn, nєZ.
2cos(x)+1=0
cos(x)=-1/2
x1=2π/3 +2πn, nєZ,
x2=4π/3+2πn, nєZ.
___o_____o_____o_____o____
..+..π/2...-...2π/3..+..4π/3...-...3π/2..+.
xє(π/2+2πn;2π/3+2πn)U(4π/3+2πn;3π/2+2πn), nєZ.