1. Let $x$ be the number. $4(6x + 7) = 30(-x) + 10$ $24x + 28 = -30x + 10$ $24x + 30x + 28 = 10$ $54x + 28 = 10$ $54x = 10 - 28$ $54x = -18$ $x = \frac{-18}{54} = -\frac{1}{3}$ 2. $\text{Overall average} = \frac{5 \times 6.5 + 15 \times 4.5}{5 + 15}$ $= \frac{32.5 + 67.5}{20} = \frac{100}{20} = 5$ 3. $5 \frac{1}{8} = \frac{41}{8}, 3 \frac{7}{16} = \frac{55}{16}$ $\frac{41}{8} x = \frac{55}{16}$ $x = \frac{55}{16} \times \frac{8}{41} = \frac{55}{82}$ 4. $\frac{0.00008}{0.004} = 0.02$ 5. $3 \frac{2}{5} = \frac{17}{5}, -2 \frac{3}{8} = -\frac{19}{8}$ $\frac{17}{5} p + \frac{1}{2} = -\frac{19}{8}$ $\frac{17}{5} p = -\frac{19}{8} - \frac{1}{2} = -\frac{23}{8}$ $p = -\frac{23}{8} \times \frac{5}{17} = -\frac{115}{136}$ 6. $-(-k) = k$ $-(-2)(2k - 5) = 2(2k - 5) = 4k - 10$ $k + 4k - 10 + 7 = -2k - 4$ $5k - 3 = -2k - 4$ $7k - 3 = -4$ $7k = -1$ $k = -\frac{1}{7}$ 7. $2 < x \leq 4$ 8. $K = \{0,1,5,0,7,5,2,0,7\} = \{0,1,2,5,7\}$ 9. $7 \in B$? No $3 \notin A$? No $0 \in B$? Yes 10. $x = -19 - 6y$ $2x + 3y = -11$ $2(-19 - 6y) + 3y = -11$ $-38 - 12y + 3y = -11$ $-38 - 9y = -11$ $-9y = 27$ $y = -3$ $x = -19 - 6(-3) = -1$ 11. $2x - 3y = 5$ $x = -2y - 8$ $2(-2y - 8) - 3y = 5$ $-4y - 16 - 3y = 5$ $-7y - 16 = 5$ $-7y = 21$ $y = -3$ $x = -2(-3) - 8 = -2$ 12. $y = 4x + 9$ $3x + y = -12$ $3x + 4x + 9 = -12$ $7x + 9 = -12$ $7x = -21$ $x = -3$ $y = 4(-3) + 9 = -3$ 13. $(5 + 3x)(8 - 2x) = 40 - 10x + 24x - 6x^2 = 40 + 14x - 6x^2$ 14. $(4x + 2)^2 = 16x^2 + 16x + 4$ 15. $y = -3$ $3y + x = -9 \Rightarrow y = \frac{-x - 9}{3}$ 16. $\frac{x}{\frac{1}{a+b}} = x(a+b)$ 17. $\frac{1}{\frac{a+b}{x}} = \frac{x}{a+b}$ 18. $\frac{a}{b} \div \frac{1}{x} = \frac{ax}{b}$ 19. $(a+b) \div \frac{1}{x} = x(a+b)$ 20. $\frac{x}{y} + \frac{1}{y+1}$ 21. $1 + \frac{x}{y} = \frac{x + y}{y}$ 22. $y - \frac{1}{y} = \frac{y^2 - 1}{y}$ 23. $10x^3 y^2 z - 5x^3 y^2 z^5 - 10x^2 y^4 z^4 = 5x^2 y^2 z (2x - x z^4 - 2 y^2 z^3)$ 24. $\frac{(x^2 y^0 m)(m^{-2} y)}{m^2 (m y^{-2})} = \frac{x^2 y m^{-1}}{m^3 y^{-2}} = x^2 m^{-4} y^3 = \frac{x^2 y^3}{m^4}$ 25. $\frac{(x^0 y)^{-2} y^5 x}{x^2 x^{-5} y y^{-3}} = \frac{y^{-2} y^5 x}{x^{-3} y^{-2}} = x^4 y^5$ 26. $\frac{(x y^{-2})^{-3}}{m^2} \times \frac{(y^{-2})^{0}}{(2 x^{7})^{-3}} = \frac{x^{-3} y^{6}}{m^2} \times 8 x^{21} = \frac{8 x^{18} y^{6}}{m^2}$ 27. $-2 \{[(3 - 5) - (2^{0} - 6) - 2] - [(4 - 3) - 2(-3)]\} + \sqrt[3]{-8}$ $= -2 \{[-2 - (1 - 6) - 2] - [1 + 6]\} - 2$ $= -2 \{[-2 + 5 - 2] - 7\} - 2$ $= -2 (1 - 7) - 2 = -2 (-6) - 2 = 12 - 2 = 10$