1. Find four consecutive integers such that if the sum of the first and third is increased by 10, the result is 6 greater than 4 times the fourth. Let the four consecutive integers be $x$, $x+1$, $x+2$, and $x+3$. Equation: $ (x + (x+2)) + 10 = 4(x+3) + 6 $ Simplify: $ 2x + 2 + 10 = 4x + 12 + 6 $ $ 2x + 12 = 4x + 18 $ Subtract $2x$ both sides: $ \cancel{2x} + 12 = \cancel{2x} + 18 + 2x $ $ 12 = 2x + 18 $ Subtract 18 both sides: $ 12 - 18 = 2x + \cancel{18} - 18 $ $ -6 = 2x $ Divide both sides by 2: $ \frac{-6}{\cancel{2}} = \frac{2x}{\cancel{2}} $ $ x = -3 $ The integers are $-3$, $-2$, $-1$, $0$. 2. The new hog food supplement increased the weight gain by $\frac{2}{5}$. If the weight gain used to be 300 pounds, what was the new weight gain? Increase: $ 300 \times \frac{2}{5} = 120 $ New weight gain: $ 300 + 120 = 420 $ 3. When the leprechauns ran into the forest, the number of little people present was decreased by 35 percent. If 105 ran into the forest, how many were left? Let total number be $x$. Decrease: $ 0.35x $ Remaining: $ x - 0.35x = 0.65x $ Given $ 0.35x = 105 $ Solve for $x$: $ x = \frac{105}{0.35} = 300 $ Left: $ 0.65 \times 300 = 195 $ 4. What fraction of $7 \frac{2}{5}$ is $49 \frac{1}{3}$? Convert to improper fractions: $7 \frac{2}{5} = \frac{37}{5}$ $49 \frac{1}{3} = \frac{148}{3}$ Fraction: $ \frac{\frac{148}{3}}{\frac{37}{5}} = \frac{148}{3} \times \frac{5}{37} = \frac{148 \times 5}{3 \times 37} = \frac{740}{111} $ 5. $-\frac{\sqrt{3}}{2}$ belongs to which sets of numbers? It is a real number, irrational number, and a real negative number. 6. Simplify: $$\frac{\frac{x}{y} - 1}{\frac{x}{y} + m}$$ Rewrite numerator and denominator: Numerator: $\frac{x}{y} - 1 = \frac{x - y}{y}$ Denominator: $\frac{x}{y} + m = \frac{x + my}{y}$ Expression: $$\frac{\frac{x - y}{y}}{\frac{x + my}{y}} = \frac{x - y}{y} \times \frac{y}{x + my} = \frac{x - y}{x + my}$$ 7. Simplify: $$\frac{x^2 (2y^{-2})^{-3}}{(4x^2)^{-2}}$$ Simplify powers: $ (2y^{-2})^{-3} = 2^{-3} y^{6} = \frac{y^6}{8} $ $ (4x^2)^{-2} = 4^{-2} x^{-4} = \frac{1}{16 x^4} $ Expression: $$\frac{x^2 \times \frac{y^6}{8}}{\frac{1}{16 x^4}} = x^2 \times \frac{y^6}{8} \times 16 x^4 = \frac{16}{8} x^{2+4} y^6 = 2 x^6 y^6$$ 8. Simplify: $$\frac{1}{-3^{-2}} + (-3^0)(-3 - 5)$$ Calculate terms: $-3^{-2} = -\frac{1}{9}$ $\frac{1}{-3^{-2}} = \frac{1}{-\frac{1}{9}} = -9$ $-3^0 = -1$ $-3 - 5 = -8$ Expression: $-9 + (-1)(-8) = -9 + 8 = -1$ 9. Simplify by adding like terms: $$\frac{ax^{-4}}{(x^{-2})^2} + \frac{3a^{-2}a^3}{a^0} - \frac{6a^5}{(a^{-2})^{-2}} + 3a$$ Simplify each term: $ (x^{-2})^2 = x^{-4} $ First term: $ \frac{ax^{-4}}{x^{-4}} = a $ Second term: $ \frac{3a^{-2}a^3}{1} = 3a^{1} = 3a $ Third term: $ (a^{-2})^{-2} = a^{4} $ $ \frac{6a^5}{a^4} = 6a^{1} = 6a $ Fourth term: $3a$ Sum: $a + 3a - 6a + 3a = (1 + 3 - 6 + 3)a = 1a = a$ 10. Simplify: $$\frac{(0.003 \times 10^7)(700000)}{(5000)(0.0021 \times 10^{-6})}$$ Calculate numerator: $0.003 \times 10^7 = 30000$ $30000 \times 700000 = 2.1 \times 10^{10}$ Calculate denominator: $5000 \times 0.0021 \times 10^{-6} = 5000 \times 2.1 \times 10^{-3} \times 10^{-6} = 5000 \times 2.1 \times 10^{-9} = 1.05 \times 10^{-5}$ Expression: $$\frac{2.1 \times 10^{10}}{1.05 \times 10^{-5}} = 2 \times 10^{15}$$ 11. Simplify: $$\frac{(0.0007 \times 10^{-10})(4000 \times 10^5)}{(0.0004)(7000)}$$ Calculate numerator: $0.0007 \times 10^{-10} = 7 \times 10^{-14}$ $4000 \times 10^5 = 4 \times 10^8$ Numerator: $7 \times 10^{-14} \times 4 \times 10^8 = 28 \times 10^{-6} = 2.8 \times 10^{-5}$ Denominator: $0.0004 \times 7000 = 2.8$ Expression: $$\frac{2.8 \times 10^{-5}}{2.8} = 10^{-5}$$ 12. Four-fifths of the delegates crowded into the convention hall. If 140 could not get in, how many attended the convention? Let total delegates be $x$. Delegates inside: $\frac{4}{5}x$ Delegates outside: $\frac{1}{5}x = 140$ Solve for $x$: $x = 140 \times 5 = 700$ Attended: $\frac{4}{5} \times 700 = 560$ 13. Evaluate: $$\frac{\sqrt[3]{x + 2} + 11}{3}$$ Given: $$\frac{x - 22}{5} = 2^3 = 8$$ Solve for $x$: $x - 22 = 40$ $x = 62$ Evaluate numerator: $\sqrt[3]{62 + 2} + 11 = \sqrt[3]{64} + 11 = 4 + 11 = 15$ Expression: $\frac{15}{3} = 5$ 14. Use substitution to solve: $x + 5y = 17$ $2x - 4y = -8$ From first: $x = 17 - 5y$ Substitute into second: $2(17 - 5y) - 4y = -8$ $34 - 10y - 4y = -8$ $34 - 14y = -8$ $-14y = -42$ $y = 3$ $x = 17 - 5(3) = 17 - 15 = 2$ Solution: $(x,y) = (2,3)$ 15. Use elimination to solve: $N_N + N_D = 30$ $5N_N + 10N_D = 250$ Multiply first by 5: $5N_N + 5N_D = 150$ Subtract from second: $(5N_N + 10N_D) - (5N_N + 5N_D) = 250 - 150$ $5N_D = 100$ $N_D = 20$ $N_N = 30 - 20 = 10$ Solution: $N_N = 10$, $N_D = 20$ 16. Solve: $\frac{x}{3} + \frac{5x + 3}{2} = 5$ Multiply both sides by 6: $2x + 3(5x + 3) = 30$ $2x + 15x + 9 = 30$ $17x + 9 = 30$ $17x = 21$ $x = \frac{21}{17}$ 17. Solve: $\frac{y + 3}{2} - \frac{4y}{3} = \frac{1}{6}$ Multiply both sides by 6: $3(y + 3) - 8y = 1$ $3y + 9 - 8y = 1$ $-5y + 9 = 1$ $-5y = -8$ $y = \frac{8}{5}$ 18. Expand: $$\frac{x^{-2}}{a^2 y^{-2}} \left( \frac{x^4 a^5}{y^4} - \frac{3x^{-4}}{a^{-4} y^2} \right)$$ Rewrite denominator: $a^2 y^{-2} = a^2 \times y^{-2}$ Expression inside parentheses: First term: $\frac{x^4 a^5}{y^4}$ Second term: $\frac{3x^{-4}}{a^{-4} y^2} = 3x^{-4} a^{4} y^{-2}$ Multiply entire expression: $$x^{-2} a^{-2} y^{2} \left( \frac{x^4 a^5}{y^4} - 3x^{-4} a^{4} y^{-2} \right)$$ Distribute: First term: $x^{-2} a^{-2} y^{2} \times \frac{x^4 a^5}{y^4} = x^{-2+4} a^{-2+5} y^{2-4} = x^{2} a^{3} y^{-2}$ Second term: $x^{-2} a^{-2} y^{2} \times (-3x^{-4} a^{4} y^{-2}) = -3 x^{-2-4} a^{-2+4} y^{2-2} = -3 x^{-6} a^{2} y^{0} = -3 x^{-6} a^{2}$ Final expression: $$x^{2} a^{3} y^{-2} - 3 x^{-6} a^{2}$$ 19. Simplify: $4 \sqrt{28} - 3 \sqrt{63} + \sqrt{175}$ Simplify radicals: $\sqrt{28} = \sqrt{4 \times 7} = 2 \sqrt{7}$ $\sqrt{63} = \sqrt{9 \times 7} = 3 \sqrt{7}$ $\sqrt{175} = \sqrt{25 \times 7} = 5 \sqrt{7}$ Expression: $4 \times 2 \sqrt{7} - 3 \times 3 \sqrt{7} + 5 \sqrt{7} = 8 \sqrt{7} - 9 \sqrt{7} + 5 \sqrt{7} = (8 - 9 + 5) \sqrt{7} = 4 \sqrt{7}$ 20. Given: $R_G T_G + R_B T_B = 120$, $R_G = 4$, $R_B = 10$, $T_G = T_B + 2$ Substitute: $4(T_B + 2) + 10 T_B = 120$ $4 T_B + 8 + 10 T_B = 120$ $14 T_B + 8 = 120$ $14 T_B = 112$ $T_B = 8$ $T_G = 8 + 2 = 10$ 21. Given: $R_K T_K = R_N T_N$, $R_K = 6$, $R_N = 3$, $T_K = T_N - 8$ Substitute: $6 (T_N - 8) = 3 T_N$ $6 T_N - 48 = 3 T_N$ $6 T_N - 3 T_N = 48$ $3 T_N = 48$ $T_N = 16$ $T_K = 16 - 8 = 8$ 22. Solve by graphing: $y = x - 6$ $y = -x$ Set equal: $x - 6 = -x$ $2x = 6$ $x = 3$ $y = -3$ Solution: $(3, -3)$ 23. Solve by graphing: $y = x + 1$ $y = -x - 1$ Set equal: $x + 1 = -x - 1$ $2x = -2$ $x = -1$ $y = -1 + 1 = 0$ Solution: $(-1, 0)$ 24. Factor: $ax^2 + 6a - 7ax$ Group terms: $ax^2 - 7ax + 6a$ Factor $a$: $a(x^2 - 7x + 6)$ Factor quadratic: $a(x - 6)(x - 1)$ 25. Factor: $-mx^2 - 8m - 6mx$ Rewrite: $-m x^2 - 6 m x - 8 m$ Factor $-m$: $-m (x^2 + 6x + 8)$ Factor quadratic: $-m (x + 4)(x + 2)$ 26. Factor: $mx^2 - 9ma^2$ Factor $m$: $m(x^2 - 9a^2)$ Difference of squares: $m(x - 3a)(x + 3a)$ 27. Factor: $-k^2 + 4m^2 x^2$ Rewrite: $4 m^2 x^2 - k^2$ Difference of squares: $(2 m x - k)(2 m x + k)$