1. Problem: Find three consecutive integers where 5 times the sum of the first two is 7 more than the opposite of the third. Let the integers be $x$, $x+1$, $x+2$. Equation: $5(x + (x+1)) = -(x+2) + 7$ Simplify: $5(2x+1) = -x - 2 + 7$ $10x + 5 = -x + 5$ Add $x$ to both sides: $10x + x + 5 = 5$ $11x + 5 = 5$ Subtract 5: $11x = 0$ $x = 0$ Integers: $0, 1, 2$ 2. Problem: Find four consecutive even integers where 3 times the sum of the first two is 18 more than 5 times the third. Let the integers be $x$, $x+2$, $x+4$, $x+6$. Equation: $3(x + x+2) = 5(x+4) + 18$ Simplify: $3(2x+2) = 5x + 20 + 18$ $6x + 6 = 5x + 38$ Subtract $5x$: $x + 6 = 38$ $x = 32$ Integers: $32, 34, 36, 38$ 3. Problem: Price reduced by 4% to sell for 3440. Find original price and reduction. Let original price be $P$. Equation: $P - 0.04P = 3440$ $0.96P = 3440$ $P = \frac{3440}{0.96} = 3583.33$ Reduction: $3583.33 - 3440 = 143.33$ 4. Problem: Flood tide rise increased inundated land by 270%. Former inundation 1400 acres. Increase: $270\% = 2.7$ times original. New inundation: $1400 + 2.7 \times 1400 = 1400 + 3780 = 5180$ acres. 5. Problem: Ezekiel has 7452 jeroboams, 1.62 times the next country. Let next country have $x$ jeroboams. $1.62x = 7452$ $x = \frac{7452}{1.62} = 4600$ 6. Problem: $\frac{7\sqrt{2}}{3}$ belongs to which sets? Answer: Rational multiples of irrational numbers, so it belongs to the set of real numbers and irrational numbers. 7. Simplify: $-2(4 - 1)(-1 - 2^0) + |-3 + 5|$ Calculate powers: $2^0 = 1$ Expression: $-2(3)(-1 - 1) + |2| = -2(3)(-2) + 2 = 12 + 2 = 14$ 8. What fraction of $2 \frac{1}{4}$ is $7 \frac{3}{8}$? Convert to improper fractions: $2 \frac{1}{4} = \frac{9}{4}$, $7 \frac{3}{8} = \frac{59}{8}$ Fraction: $\frac{59/8}{9/4} = \frac{59}{8} \times \frac{4}{9} = \frac{236}{72} = \frac{59}{18}$ 9. Solve: $2.2x - 0.1x + 0.02x = -2 - 0.32x$ Combine like terms: $2.12x = -2 - 0.32x$ Add $0.32x$: $2.44x = -2$ $x = \frac{-2}{2.44} = -0.82$ 10. Evaluate: $\frac{\sqrt[3]{x} + 7}{3}$ if $x = -50$ Calculate cube root: $\sqrt[3]{-50} = -3.684$ Expression: $\frac{-3.684 + 7}{3} = \frac{3.316}{3} = 1.105$ 11. Find equations of lines (bottom-left grid): Line 1 passes through (0,1) and (2,3): slope $m = \frac{3-1}{2-0} = 1$ Equation: $y = x + 1$ Line 2 passes through (0,3) and (2,1): slope $m = \frac{1-3}{2-0} = -1$ Equation: $y = -x + 3$ 12. Find equations of lines (bottom-left grid): Line 1 passes through (0,0) and (3,3): slope $m=1$ Equation: $y = x$ Line 2 passes through (0,3) and (3,0): slope $m = -1$ Equation: $y = -x + 3$ 14. Simplify: $$\frac{(0.0056 \times 10^{-3})(100000 \times 10^{-14})}{8000 \times 10^{15}}$$ Calculate numerator: $0.0056 \times 10^{-3} = 5.6 \times 10^{-6}$ $100000 \times 10^{-14} = 10^5 \times 10^{-14} = 10^{-9}$ Numerator: $5.6 \times 10^{-6} \times 10^{-9} = 5.6 \times 10^{-15}$ Denominator: $8000 \times 10^{15} = 8 \times 10^3 \times 10^{15} = 8 \times 10^{18}$ Fraction: $\frac{5.6 \times 10^{-15}}{8 \times 10^{18}} = 0.7 \times 10^{-33} = 7 \times 10^{-34}$ 15. Simplify: $\frac{x^2 + y}{a - \frac{x}{y}}$ Rewrite denominator: $a - \frac{x}{y} = \frac{ay - x}{y}$ Expression: $\frac{x^2 + y}{\frac{ay - x}{y}} = (x^2 + y) \times \frac{y}{ay - x} = \frac{y(x^2 + y)}{ay - x}$ 16. Add: $\frac{4}{x + y} + \frac{6}{x} - \frac{4}{ax}$ Common denominator: $a x (x + y)$ Rewrite terms: $\frac{4 a x}{a x (x + y)} + \frac{6 a (x + y)}{a x (x + y)} - \frac{4 (x + y)}{a x (x + y)}$ Sum numerator: $4 a x + 6 a (x + y) - 4 (x + y)$ Final: $\frac{4 a x + 6 a (x + y) - 4 (x + y)}{a x (x + y)}$ 17. Expand: $\frac{x^2 a}{y^2} \cdot \left( \frac{a^4 y^{-2}}{x} - \frac{3 x^2 a}{y^2} \right)$ Distribute: $\frac{x^2 a}{y^2} \cdot \frac{a^4 y^{-2}}{x} = \frac{x^2 a a^4 y^{-2}}{y^2 x} = \frac{x a^5 y^{-2}}{y^2} = x a^5 y^{-4}$ $\frac{x^2 a}{y^2} \cdot \left(- \frac{3 x^2 a}{y^2} \right) = - \frac{3 x^4 a^2}{y^4}$ Sum: $x a^5 y^{-4} - 3 x^4 a^2 y^{-4} = y^{-4} (x a^5 - 3 x^4 a^2)$ 18. Simplify by adding like terms: $a^2 x y - 3 a^2 x / y^{-1} + 4 x / (y^{-1} a^2) + 5 x^{-1} y / a^2$ Rewrite powers: $-3 a^2 x / y^{-1} = -3 a^2 x y$ $4 x / (y^{-1} a^2) = 4 x y / a^2$ $5 x^{-1} y / a^2 = 5 y / (a^2 x)$ Group terms: $a^2 x y - 3 a^2 x y = -2 a^2 x y$ Final: $-2 a^2 x y + 4 x y / a^2 + 5 y / (a^2 x)$ 19. Solve by graphing: $y = x + 4$ $y = -x + 2$ Set equal: $x + 4 = -x + 2$ $2x = -2$ $x = -1$ $y = -1 + 4 = 3$ Solution: $(-1, 3)$ 20. Use substitution: $5N_M + 10N_D = 450$ $N_D = N_N + 30$ Substitute $N_D$: $5N_M + 10(N_N + 30) = 450$ $5N_M + 10N_N + 300 = 450$ $5N_M + 10N_N = 150$ No value for $N_M$ or $N_N$ given, solution in terms of variables. 21. Use elimination: $5x - 2y = 7$ $4x + y = 3$ Multiply second by 2: $8x + 2y = 6$ Add to first: $5x - 2y + 8x + 2y = 7 + 6$ $13x = 13$ $x = 1$ Substitute: $4(1) + y = 3$ $y = -1$ Solution: $(1, -1)$ 22. Solve: $\frac{p}{4} - \frac{p+2}{6} = -4$ Common denominator 12: $3p - 2(p+2) = -48$ $3p - 2p - 4 = -48$ $p - 4 = -48$ $p = -44$ 23. Solve: $k + \frac{2}{5} - \frac{k}{10} = \frac{3}{20}$ Multiply all by 20: $20k + 8 - 2k = 3$ $18k + 8 = 3$ $18k = -5$ $k = -\frac{5}{18}$ 24. Simplify: $3\sqrt{72} - 14\sqrt{18} + 6\sqrt{8}$ Simplify radicals: $\sqrt{72} = 6\sqrt{2}$ $\sqrt{18} = 3\sqrt{2}$ $\sqrt{8} = 2\sqrt{2}$ Expression: $3(6\sqrt{2}) - 14(3\sqrt{2}) + 6(2\sqrt{2}) = 18\sqrt{2} - 42\sqrt{2} + 12\sqrt{2} = (18 - 42 + 12)\sqrt{2} = -12\sqrt{2}$ 25. Given: $R_T T_T = R_B T_B$, $R_T = 80$, $R_B = 20$, $T_B = T_T + 18$ Equation: $80 T_T = 20 (T_T + 18)$ $80 T_T = 20 T_T + 360$ $60 T_T = 360$ $T_T = 6$ $T_B = 6 + 18 = 24$ 26. Given: $R_L T_L = R_S T_S$, $R_L = 120$, $R_S = 280$, $T_S = 20 - T_L$ Equation: $120 T_L = 280 (20 - T_L)$ $120 T_L = 5600 - 280 T_L$ $400 T_L = 5600$ $T_L = 14$ $T_S = 20 - 14 = 6$ 27. Factor: $4x^2 + 40x + 100$ Common factor 4: $4(x^2 + 10x + 25)$ Factor quadratic: $4(x + 5)^2$ 28. Factor: $-x^3 - 30x - 11x^2$ Rewrite: $-x^3 - 11x^2 - 30x$ Common factor $-x$: $-x(x^2 + 11x + 30)$ Factor quadratic: $-x(x + 5)(x + 6)$ 29. Factor: $ax^2 - 35a + 2ax$ Common factor $a$: $a(x^2 - 35 + 2x)$ Rewrite quadratic: $a(x^2 + 2x - 35)$ Factor quadratic: $a(x + 7)(x - 5)$