1. **Problem:** Six times a number increased by 7, then multiplied by 4, equals 10 more than 30 times the opposite of the number. Find the number. 2. **Problem:** Find the overall average of two groups: first group of 5 with average 6.5, second group of 15 with average 4.5. 3. **Problem:** Find the number such that 5 1/8 of it equals 3/16. 4. **Problem:** What decimal part of 0.004 is 0.00008? 5. **Problem:** Solve for p: $3 \frac{5}{2} p + \frac{1}{2} = -2 \frac{3}{8}$. 6. **Problem:** Solve for k: $-(-k) - (-2)(2k - 5) + 7 = -(2k + 4)$. 7. **Problem:** Graph $2 < x \leq 4$ on a number line. 8. **Problem:** Designate set K = {0, 1, 5, 0, 7, 5, 2, 0, 7} using set notation. 9. **Problem:** Given $A = \{1, 2, 3\}$ and $B = \{2, 3, 4, 0\}$, determine truth of: (a) $7 \in B$ (b) $3 \notin A$ (c) $0 \in B$ 10. **Problem:** Solve systems by substitution: (a) $\begin{cases} x = -19 - 6y \\ 2x + 3y = -11 \end{cases}$ (b) $\begin{cases} 2x - 3y = 5 \\ x = -2y - 8 \end{cases}$ (c) $\begin{cases} y = 4x + 9 \\ 3x + y = -12 \end{cases}$ 11. **Problem:** Multiply: (a) $(5 + 3x)(8 - 2x)$ (b) $(4x + 2)^2$ 12. **Problem:** Graph on rectangular coordinate system: (a) $y = -3$ (b) $3y + x = -9$ 13. **Problem:** Simplify: (a) $\frac{x}{\frac{1}{a+b}}$ (b) $\frac{\frac{1}{a+b}}{x}$ (c) $\frac{\frac{a}{b}}{\frac{1}{x}}$ (d) $\frac{\frac{a+b}{x}}{\frac{1}{x}}$ 14. **Problem:** Add: (a) $\frac{x}{y} + \frac{1}{y+1}$ (b) $1 + \frac{x}{y}$ (c) $y - \frac{1}{y}$ 15. **Problem:** Factor: $10x^2 y^5 z - 5x^5 y^2 z^5 - 10x^4 y^1 z^{-4}$ 16. **Problem:** Simplify: (a) $\frac{(x^2 y^0 m)(m^{-2} y)}{m^2 (m y^{-2})}$ (b) $\frac{(x^0 y^2)^{-2} y^5 x}{x^2 x^{-5} y y^{-3}}$ (c) $\frac{\frac{(x y^{-2})^{-3} (y^{-2})^0}{m^2}}{(2x^2)^{-3}}$ 17. **Problem:** Simplify expression: $-2\{[(3-5) - (2^0 - 6) - 2] - [(4 - 3) - 2(-3)]\} + \sqrt[3]{-8}$ --- **Solutions:** 1. Let the number be $x$. Expression: $4(6x + 7) = 30(-x) + 10$ Step 1: Expand left side: $$4(6x + 7) = 24x + 28$$ Step 2: Write right side: $$30(-x) + 10 = -30x + 10$$ Step 3: Equation: $$24x + 28 = -30x + 10$$ Step 4: Add $30x$ both sides: $$24x + 30x + 28 = 10$$ $$54x + 28 = 10$$ Step 5: Subtract 28 both sides: $$54x = 10 - 28$$ $$54x = -18$$ Step 6: Divide both sides by 54: $$x = \frac{\cancel{54}x}{\cancel{54}} = \frac{-18}{54} = -\frac{1}{3}$$ **Answer:** $x = -\frac{1}{3}$ 2. Total sum first group: $5 \times 6.5 = 32.5$ Total sum second group: $15 \times 4.5 = 67.5$ Total sum combined: $32.5 + 67.5 = 100$ Total number combined: $5 + 15 = 20$ Overall average: $$\frac{100}{20} = 5$$ **Answer:** Overall average is 5 3. Let the number be $n$. Convert mixed number to improper fraction: $$5 \frac{1}{8} = \frac{41}{8}$$ Equation: $$\frac{41}{8} n = \frac{3}{16}$$ Multiply both sides by reciprocal $\frac{8}{41}$: $$n = \frac{3}{16} \times \frac{8}{41} = \frac{3 \times 8}{16 \times 41} = \frac{24}{656}$$ Simplify numerator and denominator by 8: $$n = \frac{3}{82}$$ **Answer:** $n = \frac{3}{82}$ 4. Find decimal part $x$ such that: $$x \times 0.004 = 0.00008$$ Divide both sides by 0.004: $$x = \frac{0.00008}{0.004} = 0.02$$ **Answer:** Decimal part is 0.02 5. Rewrite mixed numbers as improper fractions: $$3 \frac{5}{2} = 3 + \frac{5}{2} = \frac{6}{2} + \frac{5}{2} = \frac{11}{2}$$ $$-2 \frac{3}{8} = -\left(2 + \frac{3}{8}\right) = -\frac{19}{8}$$ Equation: $$\frac{11}{2} p + \frac{1}{2} = -\frac{19}{8}$$ Step 1: Subtract $\frac{1}{2}$ both sides: $$\frac{11}{2} p = -\frac{19}{8} - \frac{1}{2} = -\frac{19}{8} - \frac{4}{8} = -\frac{23}{8}$$ Step 2: Multiply both sides by reciprocal $\frac{2}{11}$: $$p = -\frac{23}{8} \times \frac{2}{11} = -\frac{46}{88}$$ Simplify numerator and denominator by 2: $$p = -\frac{23}{44}$$ **Answer:** $p = -\frac{23}{44}$ 6. Equation: $$-(-k) - (-2)(2k - 5) + 7 = -(2k + 4)$$ Step 1: Simplify left side: $$k + 2(2k - 5) + 7 = -2k - 4$$ Step 2: Expand: $$k + 4k - 10 + 7 = -2k - 4$$ $$5k - 3 = -2k - 4$$ Step 3: Add $2k$ both sides: $$5k + 2k - 3 = -4$$ $$7k - 3 = -4$$ Step 4: Add 3 both sides: $$7k = -1$$ Step 5: Divide both sides by 7: $$k = -\frac{1}{7}$$ **Answer:** $k = -\frac{1}{7}$ 7. Graph $2 < x \leq 4$ means all $x$ values greater than 2 but less than or equal to 4. On number line: - Open circle at 2 (not included) - Closed circle at 4 (included) - Shade region between 2 and 4 8. Set K = {0, 1, 5, 0, 7, 5, 2, 0, 7} Remove duplicates: $$K = \{0, 1, 2, 5, 7\}$$ 9. (a) $7 \in B$? No, 7 is not in $B = \{2, 3, 4, 0\}$, so False. (b) $3 \notin A$? $3 \in A = \{1, 2, 3\}$, so False. (c) $0 \in B$? Yes, 0 is in $B$, so True. 10. (a) $x = -19 - 6y$ Substitute into second equation: $$2(-19 - 6y) + 3y = -11$$ $$-38 - 12y + 3y = -11$$ $$-38 - 9y = -11$$ Add 38 both sides: $$-9y = 27$$ Divide by -9: $$y = -3$$ Find $x$: $$x = -19 - 6(-3) = -19 + 18 = -1$$ **Answer:** $(x, y) = (-1, -3)$ (b) $x = -2y - 8$ Substitute into first equation: $$2(-2y - 8) - 3y = 5$$ $$-4y - 16 - 3y = 5$$ $$-7y - 16 = 5$$ Add 16 both sides: $$-7y = 21$$ Divide by -7: $$y = -3$$ Find $x$: $$x = -2(-3) - 8 = 6 - 8 = -2$$ **Answer:** $(x, y) = (-2, -3)$ (c) Substitute $y = 4x + 9$ into second equation: $$3x + (4x + 9) = -12$$ $$7x + 9 = -12$$ Subtract 9: $$7x = -21$$ Divide by 7: $$x = -3$$ Find $y$: $$y = 4(-3) + 9 = -12 + 9 = -3$$ **Answer:** $(x, y) = (-3, -3)$ 11. (a) Multiply: $$(5 + 3x)(8 - 2x) = 5 \times 8 + 5 \times (-2x) + 3x \times 8 + 3x \times (-2x)$$ $$= 40 - 10x + 24x - 6x^2 = 40 + 14x - 6x^2$$ (b) Square: $$(4x + 2)^2 = (4x)^2 + 2 \times 4x \times 2 + 2^2 = 16x^2 + 16x + 4$$ 12. (a) Graph $y = -3$ is a horizontal line crossing y-axis at -3. (b) Rewrite $3y + x = -9$ as: $$x = -9 - 3y$$ This is a line with slope $-3$ and y-intercept at $y=0, x=-9$. 13. (a) Simplify: $$\frac{x}{\frac{1}{a+b}} = x \times (a+b) = x(a+b)$$ (b) Simplify: $$\frac{\frac{1}{a+b}}{x} = \frac{1}{x(a+b)}$$ (c) Simplify: $$\frac{\frac{a}{b}}{\frac{1}{x}} = \frac{a}{b} \times x = \frac{a x}{b}$$ (d) Simplify: $$\frac{\frac{a+b}{x}}{\frac{1}{x}} = \frac{a+b}{x} \times x = a + b$$ 14. (a) Add: $$\frac{x}{y} + \frac{1}{y+1} = \frac{x(y+1) + y}{y(y+1)} = \frac{xy + x + y}{y(y+1)}$$ (b) Add: $$1 + \frac{x}{y} = \frac{y}{y} + \frac{x}{y} = \frac{x + y}{y}$$ (c) Add: $$y - \frac{1}{y} = \frac{y^2 - 1}{y}$$ 15. Factor: $$10x^2 y^5 z - 5x^5 y^2 z^5 - 10x^4 y z^{-4}$$ Common factors: - Coefficient: 5 - $x^2$ (lowest power) - $y$ (lowest power is 1) - $z^{-4}$ (lowest power) Extract: $$5 x^2 y z^{-4} (2 y^4 z^5 - x^3 y z^9 - 2 x^2)$$ 16. (a) Simplify: $$(x^2 y^0 m)(m^{-2} y) = x^2 m^{1 - 2} y^{0 + 1} = x^2 m^{-1} y$$ Denominator: $$m^2 (m y^{-2}) = m^{2 + 1} y^{-2} = m^3 y^{-2}$$ Fraction: $$\frac{x^2 m^{-1} y}{m^3 y^{-2}} = x^2 m^{-1 - 3} y^{1 - (-2)} = x^2 m^{-4} y^{3} = \frac{x^2 y^3}{m^4}$$ (b) Simplify numerator: $$(x^0 y^2)^{-2} y^5 x = y^{-4} y^5 x = y^{1} x = x y$$ Denominator: $$x^2 x^{-5} y y^{-3} = x^{-3} y^{-2}$$ Fraction: $$\frac{x y}{x^{-3} y^{-2}} = x^{1 - (-3)} y^{1 - (-2)} = x^{4} y^{3}$$ (c) Simplify numerator: $$(x y^{-2})^{-3} (y^{-2})^0 = x^{-3} y^{6} \times 1 = x^{-3} y^{6}$$ Divide by $m^2$: $$\frac{x^{-3} y^{6}}{m^2}$$ Denominator: $$(2 x^2)^{-3} = 2^{-3} x^{-6} = \frac{1}{8} x^{-6}$$ Fraction: $$\frac{\frac{x^{-3} y^{6}}{m^2}}{\frac{1}{8} x^{-6}} = \frac{x^{-3} y^{6}}{m^2} \times \frac{8 x^{6}}{1} = \frac{8 x^{3} y^{6}}{m^2}$$ 17. Simplify expression: $$-2\{[(3-5) - (2^0 - 6) - 2] - [(4 - 3) - 2(-3)]\} + \sqrt[3]{-8}$$ Calculate inside brackets: $$(3-5) = -2$$ $$(2^0 - 6) = 1 - 6 = -5$$ First bracket: $$-2 - (-5) - 2 = -2 + 5 - 2 = 1$$ Second bracket: $$(4 - 3) - 2(-3) = 1 + 6 = 7$$ Expression: $$-2 (1 - 7) + \sqrt[3]{-8} = -2 (-6) + (-2) = 12 - 2 = 10$$ **Answer:** 10