Solve the equations and inequalities.

Решение:

1. \( 7) x(x-16) - (x+8)(x-8) = \frac{19}{8} \)
   \( x^2 - 16x - (x^2 - 64) = \frac{19}{8} \)
   \( x^2 - 16x - x^2 + 64 = \frac{19}{8} \)
   \( -16x = \frac{19}{8} - 64 \)
   \( -16x = \frac{19 - 512}{8} \)
   \( -16x = -\frac{493}{8} \)
   \( x = \frac{493}{16 \times 8} = \frac{493}{128} \)
2. \( 8) -m(m+2) + (m+3)(m-3) = \frac{1}{2} \)
   \( -m^2 - 2m + m^2 - 9 = \frac{1}{2} \)
   \( -2m - 9 = \frac{1}{2} \)
   \( -2m = 9 + \frac{1}{2} \)
   \( -2m = \frac{19}{2} \)
   \( m = -\frac{19}{4} \)
3. \( 9) a(9+a) - (a+6)^2 = -\frac{1}{3} \)
   \( 9a + a^2 - (a^2 + 12a + 36) = -\frac{1}{3} \)
   \( 9a + a^2 - a^2 - 12a - 36 = -\frac{1}{3} \)
   \( -3a - 36 = -\frac{1}{3} \)
   \( -3a = 36 - \frac{1}{3} \)
   \( -3a = \frac{108 - 1}{3} = \frac{107}{3} \)
   \( a = -\frac{107}{9} \)
4. \( 10) y^2 - 4y + 4 - (y-3)^2 = \frac{13}{2} \)
   \( (y-2)^2 - (y-3)^2 = \frac{13}{2} \)
   \( (y^2 - 4y + 4) - (y^2 - 6y + 9) = \frac{13}{2} \)
   \( y^2 - 4y + 4 - y^2 + 6y - 9 = \frac{13}{2} \)
   \( 2y - 5 = \frac{13}{2} \)
   \( 2y = 5 + \frac{13}{2} = \frac{10+13}{2} = \frac{23}{2} \)
   \( y = \frac{23}{4} \)
5. \( 11) (m+1)^2 + (6-m)(6+m) = \frac{1}{2} \)
   \( m^2 + 2m + 1 + 36 - m^2 = \frac{1}{2} \)
   \( 2m + 37 = \frac{1}{2} \)
   \( 2m = \frac{1}{2} - 37 = \frac{1 - 74}{2} = -\frac{73}{2} \)
   \( m = -\frac{73}{4} \)
6. \( 12) a^2 + 10a + 25 + (5-a)(a+5) = -2.8 \)
   \( (a+5)^2 + (5-a)(a+5) = -2.8 \)
   \( (a+5) [ (a+5) + (5-a) ] = -2.8 \)
   \( (a+5) [ a+5+5-a ] = -2.8 \)
   \( (a+5) [ 10 ] = -2.8 \)
   \( 10a + 50 = -2.8 \)
   \( 10a = -52.8 \)
   \( a = -5.28 \)
7. \( 13) x(x+6) - (x+3)(x-3) = -\frac{19}{3} \)
   \( x^2 + 6x - (x^2 - 9) = -\frac{19}{3} \)
   \( x^2 + 6x - x^2 + 9 = -\frac{19}{3} \)
   \( 6x + 9 = -\frac{19}{3} \)
   \( 6x = -9 - \frac{19}{3} = \frac{-27-19}{3} = -\frac{46}{3} \)
   \( x = -\frac{46}{18} = -\frac{23}{9} \)
8. \( 14) (y-4)^2 - (6+y)(y-6) = -\frac{7}{8} \)
   \( y^2 - 8y + 16 - (36 - y^2) = -\frac{7}{8} \)
   \( y^2 - 8y + 16 - 36 + y^2 = -\frac{7}{8} \)
   \( 2y^2 - 8y - 20 = -\frac{7}{8} \)
   \( 16y^2 - 64y - 160 = -7 \)
   \( 16y^2 - 64y - 153 = 0 \)
9. \( 15) x(x+14) - (4+x)(x-7) = -\frac{3}{7} \)
   \( x^2 + 14x - (x^2 + 4x - 7x - 28) = -\frac{3}{7} \)
   \( x^2 + 14x - (x^2 - 3x - 28) = -\frac{3}{7} \)
   \( x^2 + 14x - x^2 + 3x + 28 = -\frac{3}{7} \)
   \( 17x + 28 = -\frac{3}{7} \)
   \( 17x = -28 - \frac{3}{7} = \frac{-196 - 3}{7} = -\frac{199}{7} \)
   \( x = -\frac{199}{7 \times 17} = -\frac{199}{119} \)

Ответ: 7) x = 493/128; 8) m = -19/4; 9) a = -107/9; 10) y = 23/4; 11) m = -73/4; 12) a = -5.28; 13) x = -23/9; 14) 16y^2 - 64y - 153 = 0; 15) x = -199/119.