1. Solve the inequality $6n - 12 > 30$. Start by adding 12 to both sides: $$6n - 12 + 12 > 30 + 12$$ $$6n > 42$$ Divide both sides by 6: $$\cancel{6}n > \cancel{6}7$$ $$n > 7$$ The solution is all $n$ greater than 7. 2. Write and solve the proportion: There are 16 bananas for every 3 bunches. How many bunches hold 48 bananas? Set up the proportion: $$\frac{16}{3} = \frac{48}{x}$$ Cross multiply: $$16x = 3 \times 48$$ $$16x = 144$$ Divide both sides by 16: $$\cancel{16}x = \frac{144}{\cancel{16}}$$ $$x = 9$$ So, 9 bunches hold 48 bananas. 3. For every 9 animals, there are 2 birds. There are 180 animals. How many are birds? Set up the proportion: $$\frac{2}{9} = \frac{x}{180}$$ Cross multiply: $$9x = 2 \times 180$$ $$9x = 360$$ Divide both sides by 9: $$\cancel{9}x = \frac{360}{\cancel{9}}$$ $$x = 40$$ There are 40 birds. 4. Congruent triangles: Given $\triangle QRS \cong \triangle TVU$. Complete the congruence statements: - $\triangle QRS \cong \triangle TVU$ - $\angle TVU \cong \angle QRS$ - $SQ \cong UV$ 5. Fill in missing sides for similar quadrilaterals $ABCD \sim EFGH$: Given: $$\frac{AB}{EF} = \frac{BC}{?}$$ $$\frac{FG}{BC} = \frac{HE}{?}$$ Since corresponding sides are proportional: - Missing side in denominator for first proportion is $FG$ - Missing side in denominator for second proportion is $AD$ So: $$\frac{AB}{EF} = \frac{BC}{FG}$$ $$\frac{FG}{BC} = \frac{HE}{AD}$$ 6. Solve inequality $4t + 8 < 20$. Subtract 8 from both sides: $$4t + 8 - 8 < 20 - 8$$ $$4t < 12$$ Divide both sides by 4: $$\cancel{4}t < \frac{12}{\cancel{4}}$$ $$t < 3$$ 7. Ratio problem: For every 12 children, 5 have blue swimsuits. There are 80 children. How many have blue swimsuits? Set up proportion: $$\frac{5}{12} = \frac{x}{80}$$ Cross multiply: $$12x = 5 \times 80$$ $$12x = 400$$ Divide both sides by 12: $$\cancel{12}x = \frac{400}{\cancel{12}}$$ $$x = \frac{400}{12} = 33.3$$ Approximately 33 children have blue swimsuits. 8. Parallel lines cut by a transversal: Given lines $l$ and $m$ parallel, angles numbered 1 to 8. - Corresponding angles: 1 and 5 - Alternate interior angles: 3 and 6 - Alternate exterior angles: 1 and 8 - Supplementary angles: 1 and 2 9. Measures of central tendency for data: $13, 13, 14, 15, 17, 17, 18, 18, 18, 18, 20, 21, 23, 24, 25$ - Mean: Sum all values and divide by 15. Sum = $13+13+14+15+17+17+18+18+18+18+20+21+23+24+25 = 292$ Mean = $\frac{292}{15} = 19.47$ - Median: Middle value (8th value) when ordered. Ordered data median = 18 - Mode: Most frequent value = 18 - Minimum = 13 - Maximum = 25 - Range = $25 - 13 = 12$ - Quartiles: Q1 = median of first 7 values = 15 Q2 = median = 18 Q3 = median of last 7 values = 22 - IQR = $Q3 - Q1 = 22 - 15 = 7$ 10. Measures of central tendency for data: $1, 1, 1, 2, 3, 5, 5, 5, 6, 8, 10, 11, 13, 13, 15$ - Mean: Sum = 88, count = 15 Mean = $\frac{88}{15} = 5.87$ - Median: 8th value = 5 - Mode: 1 and 5 (both appear 3 times) - Minimum = 1 - Maximum = 15 - Range = $15 - 1 = 14$ - Quartiles: Q1 = median of first 7 = 2 Q2 = median = 5 Q3 = median of last 7 = 11 - IQR = $11 - 2 = 9$