Forming Linear inequality in a word problem

Forming a linear inequality is two steps: write the quantity as an algebraic expression, then compare it to a limit with the correct sign. The sign comes straight from the wording — "at most" is $\le$, "at least" is $\ge$, "more than" is $>$, and "less than" is $<$.

Build the expression before choosing the sign. "A driver weighing $180$ pounds loads $40$-pound boxes onto an elevator that holds at most $1500$ pounds": the total weight is $180 + 40b$, and "at most $1500$" gives $180 + 40b \le 1500$.

Watch whether the boundary is included. "At most" and "at least" include the limit, so use $\le$ or $\ge$; "more than" and "less than" exclude it, so use $>$ or $<$. "Must sell more than $5000" is $s > 5000$, not $s \ge 5000$.

Percent and comparison phrasing need care. "$10\%$ greater than $x$" means $x + 0.1x = 1.1x$, so "no district has a population more than $10\%$ greater than another" becomes $y \le 1.1x$. Decide which quantity is larger before picking the direction of the sign.

A host pays a $200 flat fee plus $18 per guest and wants to spend no more than $1500. Write an inequality for the number of guests $g$.
The total cost is $18g + 200$, and "no more than $1500" gives $18g + 200 \le 1500$.

A student has $45 and earns $15 per hour, and needs at least $120. Write an inequality for the hours $h$.
The total is $15h + 45$, and "at least $120" gives $15h + 45 \ge 120$.

Company A charges $100 + 75h$ and company B charges $150 + 50h$. Write an inequality for when A is cheaper than B.
"Cheaper" means A's cost is less than B's: $100 + 75h < 150 + 50h$.