Application: Solving Real-World Problems with Multi-Step Inequalities

Property A multi step inequality models real world scenarios involving a fixed cost plus a variable rate, or comparing two different plans. 1. Define a variable (e.g., let $x$ be the number of items). 2. Translate keywords: "at least" ($\geq$), "at most" ($\leq$), "more than" ($ $), "fewer than" ($<$). 3. Build the inequality: Fixed Amount + (Rate Variable). 4. Solve and interpret the result practically (e.g., you cannot buy half a ticket).

Examples Example 1 (Budget Constraint): A gym membership costs $30 per month plus $5 per guest pass. If you want to spend at most $55 this month, how many guest passes ($g$) can you buy? Inequality: $30 + 5g \leq 55$. Solve: Subtract 30 to get $5g \leq 25$, then divide by 5 to get $g \leq 5$. You can buy at most 5 guest passes. Example 2 (Comparing Plans): Plan A charges a $15 monthly fee plus $0.10 per text. Plan B charges $0.25 per text with no fee. For how many texts ($t$) is Plan A cheaper (costs less than Plan B)? Inequality: $15 + 0.10t < 0.25t$. Solve: Subtract $0.10t$ from both sides to get $15 < 0.15t$. Divide by 0.15 to get $100 < t$ (which is $t 100$). Plan A is cheaper if you send more than 100 texts. Example 3 (Discrete Limits): A phone plan costs $40 per month plus $0.15 per text. To keep your bill strictly under $50, how many texts ($t$) can you send? Inequality: $40 + 0.15t < 50$. Solve: $0.15t < 10 \rightarrow t < 66.67$. Since you cannot send a fraction of a text, you can send at most 66 texts.

Explanation Real world math rarely requires just one step! Most scenarios involve a starting fee that happens once, plus a rate that happens repeatedly. When translating these into math, place your variable next to the rate. Multi step inequalities are incredibly powerful for making financial decisions, like figuring out exactly when a subscription plan with an upfront fee becomes a better deal than a pay as you go plan.