1. Stating the problem: "Eight is less than the sum of three-fourths times a number and two" means we want to write and solve the inequality for the number, say $x$. 2. Writing the inequality: The phrase "three-fourths times a number" is $\frac{3}{4}x$, and "sum of three-fourths times a number and two" is $\frac{3}{4}x + 2$. "Eight is less than" means $8 < \frac{3}{4}x + 2$. 3. Solve the inequality: $$8 < \frac{3}{4}x + 2$$ Subtract 2 from both sides: $$8 - 2 < \frac{3}{4}x + 2 - 2$$ $$6 < \frac{3}{4}x$$ 4. To isolate $x$, multiply both sides by the reciprocal of $\frac{3}{4}$, which is $\frac{4}{3}$: $$6 \times \frac{4}{3} < \frac{3}{4}x \times \frac{4}{3}$$ $$\cancel{6} \times \frac{4}{3} < \cancel{\frac{3}{4}}x \times \cancel{\frac{4}{3}}$$ Calculate left side: $$6 \times \frac{4}{3} = \frac{24}{3} = 8$$ So, $$8 < x$$ 5. Interpretation: The number $x$ must be greater than 8. Final answer: $$x > 8$$