1. Find four consecutive even integers such that 4 times the sum of the first and fourth is 8 greater than 12 times the third. Let the four consecutive even integers be $x$, $x+2$, $x+4$, and $x+6$. The sum of the first and fourth is $x + (x+6) = 2x + 6$. 4 times this sum is $4(2x + 6) = 8x + 24$. 12 times the third integer is $12(x+4) = 12x + 48$. According to the problem: $$8x + 24 = 12x + 48 + 8$$ Simplify the right side: $$8x + 24 = 12x + 56$$ Bring all terms to one side: $$8x + 24 - 12x - 56 = 0$$ Simplify: $$-4x - 32 = 0$$ Add 32 to both sides: $$-4x = 32$$ Divide both sides by $-4$: $$x = \frac{32}{\cancel{-4}}\cancel{-1} = -8$$ The integers are $-8$, $-6$, $-4$, and $-2$. 2. Galileo tried for a reasonable result but got $4 \frac{3}{5}$. If this was $2 \frac{3}{10}$ of his goal, what number was he trying for? Convert mixed numbers to improper fractions: $4 \frac{3}{5} = \frac{23}{5}$, $2 \frac{3}{10} = \frac{23}{10}$. Let the goal be $G$. Equation: $$\frac{23}{5} = \frac{23}{10} G$$ Divide both sides by $\frac{23}{10}$: $$G = \frac{\frac{23}{5}}{\frac{23}{10}} = \frac{23}{5} \times \frac{10}{23}$$ Cancel $23$: $$G = \cancel{\frac{23}{5}} \times \frac{10}{\cancel{23}} = \frac{10}{5} = 2$$ Galileo's goal was 2. 3. Paul wrestled at 125 pounds. Weight increased 16% in 1 year. Find new weight. New weight = original weight + 16% of original weight $$= 125 + 0.16 \times 125 = 125(1 + 0.16) = 125 \times 1.16 = 145$$ Paul wrestled at 145 pounds in ninth grade. 4. Carolyn plowed 67% of farm in 2 weeks, which is 268 acres. Find farm size. Let farm size be $F$. $$0.67 F = 268$$ Divide both sides by 0.67: $$F = \frac{268}{0.67} \approx 400$$ Farm size is approximately 400 acres. 5. Seven-eighths of workers used scurrilous language. 400 did not. Find total workers. Let total workers be $W$. Workers not using scurrilous language = $\frac{1}{8} W = 400$ Multiply both sides by 8: $$W = 400 \times 8 = 3200$$ Total workers = 3200. 6. 2 ∈ {What sets of numbers}? 2 is an integer, rational number, real number, complex number. 7. Solve $3 \frac{1}{8} x - 4 \frac{2}{5} = 7 \frac{1}{2}$. Convert mixed numbers: $3 \frac{1}{8} = \frac{25}{8}$, $4 \frac{2}{5} = \frac{22}{5}$, $7 \frac{1}{2} = \frac{15}{2}$. Equation: $$\frac{25}{8} x - \frac{22}{5} = \frac{15}{2}$$ Add $\frac{22}{5}$ to both sides: $$\frac{25}{8} x = \frac{15}{2} + \frac{22}{5}$$ Find common denominator 10: $$\frac{15}{2} = \frac{75}{10}, \quad \frac{22}{5} = \frac{44}{10}$$ Sum: $$\frac{75}{10} + \frac{44}{10} = \frac{119}{10}$$ Divide both sides by $\frac{25}{8}$: $$x = \frac{\frac{119}{10}}{\frac{25}{8}} = \frac{119}{10} \times \frac{8}{25} = \frac{119 \times 8}{10 \times 25} = \frac{952}{250}$$ Simplify numerator and denominator by 2: $$\frac{952}{250} = \frac{476}{125} = 3 \frac{101}{125}$$ 8. Solve $-[-2(x-4) - |-3|] = -2x - 8$. Calculate $|-3| = 3$. Inside bracket: $$-2(x-4) - 3 = -2x + 8 - 3 = -2x + 5$$ So left side: $$-[-2x + 5] = 2x - 5$$ Equation: $$2x - 5 = -2x - 8$$ Add $2x$ to both sides: $$4x - 5 = -8$$ Add 5 to both sides: $$4x = -3$$ Divide both sides by 4: $$x = \frac{-3}{4}$$ 9. Write $0.000135 \times 10^{-17}$ in scientific notation. $0.000135 = 1.35 \times 10^{-4}$ So: $$1.35 \times 10^{-4} \times 10^{-17} = 1.35 \times 10^{-21}$$ 10. Write $135,000 \times 10^{-17}$ in scientific notation. $135,000 = 1.35 \times 10^{5}$ So: $$1.35 \times 10^{5} \times 10^{-17} = 1.35 \times 10^{-12}$$ 11. Add $\frac{a}{xy} + \frac{4}{x(x+y)}$. Find common denominator $x y (x+y)$. Rewrite: $$\frac{a(x+y)}{x y (x+y)} + \frac{4 y}{x y (x+y)} = \frac{a(x+y) + 4 y}{x y (x+y)}$$ 12. Simplify $\frac{a^0 (2x)^{-2}}{a^2 (4 a^0)^2}$. Recall $a^0 = 1$. Rewrite numerator: $$(2x)^{-2} = \frac{1}{(2x)^2} = \frac{1}{4x^2}$$ Denominator: $$(4 a^0)^2 = 4^2 = 16$$ Expression: $$\frac{\frac{1}{4x^2}}{a^2 \times 16} = \frac{1}{4x^2} \times \frac{1}{16 a^2} = \frac{1}{64 a^2 x^2}$$ 13. Simplify by adding like terms: $$3x^2 y - \frac{4 x^{-2} y^{-2}}{x^{-4} y^{-3}} + \frac{5 x x}{y^{-1}} - \frac{3 x^2 y^2}{y^{-2}}$$ Simplify each term: Second term: $$\frac{4 x^{-2} y^{-2}}{x^{-4} y^{-3}} = 4 x^{-2 - (-4)} y^{-2 - (-3)} = 4 x^{2} y^{1} = 4 x^{2} y$$ Third term: $$\frac{5 x x}{y^{-1}} = 5 x^{2} y^{1}$$ Fourth term: $$\frac{3 x^{2} y^{2}}{y^{-2}} = 3 x^{2} y^{2 - (-2)} = 3 x^{2} y^{4}$$ Now sum: $$3 x^{2} y - 4 x^{2} y + 5 x^{2} y - 3 x^{2} y^{4} = (3 - 4 + 5) x^{2} y - 3 x^{2} y^{4} = 4 x^{2} y - 3 x^{2} y^{4}$$ 14. Solve system by substitution: $$x + 3y = 16$$ $$2x - 3y = -4$$ From first: $$x = 16 - 3y$$ Substitute into second: $$2(16 - 3y) - 3y = -4$$ $$32 - 6y - 3y = -4$$ $$32 - 9y = -4$$ $$-9y = -36$$ $$y = 4$$ Find $x$: $$x = 16 - 3(4) = 16 - 12 = 4$$ Solution: $(x,y) = (4,4)$ 15. Solve system by elimination: $$N_N + N_D = 22$$ $$5 N_N + 10 N_D = 135$$ Multiply first by 5: $$5 N_N + 5 N_D = 110$$ Subtract from second: $$(5 N_N + 10 N_D) - (5 N_N + 5 N_D) = 135 - 110$$ $$5 N_D = 25$$ $$N_D = 5$$ Find $N_N$: $$N_N + 5 = 22 \Rightarrow N_N = 17$$ 16. Expand: $$(x^{-2} y^{-2} / m^{2}) \left(x^{2} y^{2} m^{2} - \frac{3 x^{4} y^{-4}}{m^{-2}}\right)$$ Rewrite denominator in second term: $$\frac{3 x^{4} y^{-4}}{m^{-2}} = 3 x^{4} y^{-4} m^{2}$$ Expression inside parentheses: $$x^{2} y^{2} m^{2} - 3 x^{4} y^{-4} m^{2}$$ Multiply by $x^{-2} y^{-2} m^{-2}$: $$x^{-2} y^{-2} m^{-2} \times x^{2} y^{2} m^{2} - x^{-2} y^{-2} m^{-2} \times 3 x^{4} y^{-4} m^{2}$$ Simplify each term: First term: $$x^{-2+2} y^{-2+2} m^{-2+2} = x^{0} y^{0} m^{0} = 1$$ Second term: $$3 x^{-2+4} y^{-2-4} m^{-2+2} = 3 x^{2} y^{-6} m^{0} = 3 x^{2} y^{-6}$$ Result: $$1 - 3 x^{2} y^{-6}$$ 17. Simplify: $$5 \sqrt{45} - 3 \sqrt{180} + 2 \sqrt{20}$$ Simplify radicals: $$\sqrt{45} = \sqrt{9 \times 5} = 3 \sqrt{5}$$ $$\sqrt{180} = \sqrt{36 \times 5} = 6 \sqrt{5}$$ $$\sqrt{20} = \sqrt{4 \times 5} = 2 \sqrt{5}$$ Rewrite expression: $$5 \times 3 \sqrt{5} - 3 \times 6 \sqrt{5} + 2 \times 2 \sqrt{5} = 15 \sqrt{5} - 18 \sqrt{5} + 4 \sqrt{5}$$ Combine: $$(15 - 18 + 4) \sqrt{5} = 1 \sqrt{5} = \sqrt{5}$$ 18. Solve: $$\frac{x}{4} - \frac{x+2}{3} = 12$$ Multiply both sides by 12 (LCM of 4 and 3): $$3x - 4(x+2) = 144$$ $$3x - 4x - 8 = 144$$ $$-x - 8 = 144$$ $$-x = 152$$ $$x = -152$$ 19. Solve: $$\frac{2y}{4} - \frac{y}{7} = \frac{y - 3}{2}$$ Simplify first term: $$\frac{2y}{4} = \frac{y}{2}$$ Multiply entire equation by 28 (LCM of 2,7,2): $$28 \times \left(\frac{y}{2} - \frac{y}{7}\right) = 28 \times \frac{y - 3}{2}$$ $$14 y - 4 y = 14 (y - 3)$$ $$10 y = 14 y - 42$$ $$10 y - 14 y = -42$$ $$-4 y = -42$$ $$y = \frac{42}{4} = 10.5$$ 20. Solve: $$\frac{p}{6} - \frac{2p}{5} = \frac{4p - 5}{15}$$ Multiply entire equation by 30 (LCM of 6,5,15): $$30 \times \left(\frac{p}{6} - \frac{2p}{5}\right) = 30 \times \frac{4p - 5}{15}$$ $$5p - 12p = 2(4p - 5)$$ $$-7p = 8p - 10$$ $$-7p - 8p = -10$$ $$-15p = -10$$ $$p = \frac{10}{15} = \frac{2}{3}$$ 21. Given: $$R_A T_A + R_P T_P = 500$$ $$R_P = 25, T_P = 9, T_A = 5$$ Find $R_A$. Substitute known values: $$R_A \times 5 + 25 \times 9 = 500$$ $$5 R_A + 225 = 500$$ $$5 R_A = 275$$ $$R_A = \frac{275}{5} = 55$$ 22. Evaluate: $$\frac{\sqrt[3]{x}}{5}$$ Given: $$\frac{x + 20}{5} = -21$$ Multiply both sides by 5: $$x + 20 = -105$$ $$x = -125$$ Calculate cube root: $$\sqrt[3]{-125} = -5$$ Evaluate expression: $$\frac{-5}{5} = -1$$ 25. Factor: $$p^{2} - 55 - 6p$$ Rewrite: $$p^{2} - 6p - 55$$ Find factors of -55 that sum to -6: -11 and 5. Factor: $$(p - 11)(p + 5)$$ 26. Factor: $$-30 - 13x + x^{2}$$ Rewrite in standard form: $$x^{2} - 13x - 30$$ Find factors of -30 that sum to -13: -15 and 2. Factor: $$(x - 15)(x + 2)$$ 27. Factor: $$2 m^{2} - 24 m + 70$$ Factor out 2: $$2 (m^{2} - 12 m + 35)$$ Find factors of 35 that sum to -12: -7 and -5. Factor inside parentheses: $$(m - 7)(m - 5)$$ Final factorization: $$2 (m - 7)(m - 5)$$ 28. Factor: $$-x^{3} + 14 x^{2} - 40 x$$ Factor out $-x$: $$-x (x^{2} - 14 x + 40)$$ Factor quadratic: Find factors of 40 that sum to -14: -10 and -4. $$-x (x - 10)(x - 4)$$ 29. Factor: $$4 m^{2} - 49 x^{2} p^{2}$$ Recognize difference of squares: $$ (2 m)^{2} - (7 x p)^{2} = (2 m - 7 x p)(2 m + 7 x p)$$