Lesson 4: Solving Two-Step Inequalities

Property

A two-step linear inequality has the form $ax + b < c$, $ax + b \leq c$, $ax + b > c$, or $ax + b \geq c$ where $a \neq 0$. To solve a two-step inequality, use inverse operations to isolate the variable in exactly two steps, remembering to reverse the inequality sign when multiplying or dividing by a negative number.

Examples

Property

When solving inequalities containing parentheses, first apply the distributive property: $a(b + c) = ab + ac$, then solve using the standard two-step process. Remember to reverse the inequality symbol when multiplying or dividing by a negative number.

Examples

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.