Creating and Using Linear Inequalities in Context

A budget like at most 300 dollars doesn't name one answer, it names a whole set of acceptable ones: 0 dollars works, 150 works, 300 works, 301 doesn't. That's why these situations need an inequality instead of an equation. An equation pins down a single value; an inequality describes a region of values, which is what real limits actually are.

The phrase in the problem tells you which symbol to write. At least and no less than mean the value can equal the limit or go above it, so they become ≥. At most and no more than mean the value can equal the limit or go below it, so they become ≤. Strict phrases like more than, under, and fewer than exclude the boundary itself, giving you > or <. So more than 5 starts just above 5, not at it, because more than promises something bigger than 5, and 5 isn't bigger than itself.

To build the inequality, write the cost expression first, then attach the symbol the wording demands. Buying x books at 14 dollars each with no more than 120 dollars to spend gives 14x ≤ 120. Solving gives x ≤ 8.57, and since you can't buy part of a book, the answer rounds down to 8, not up. This same setup extends directly into systems of inequalities, where a second condition narrows the region further, and into interpreting graphs of inequalities, where the solution set becomes a shaded area instead of a line.