Lesson 7: Solve Multi-Step Inequalities

Property

To solve a multi-step linear inequality, follow a systematic flow:

1. Simplify each side completely (distribute and combine like terms).
2. Use the Addition or Subtraction Properties of Inequality to collect all variable terms on one side and all constant terms on the other side.
3. Use the Multiplication or Division Properties of Inequality to isolate the variable. (Remember to reverse the inequality sign if you multiply or divide by a negative number!)

Examples

• Example 1: Solve $3x + 5 > 20$.

Subtract 5 from both sides to get $3x > 15$.
Divide by 3 to get $x > 5$.

• Example 2 (Variables on both sides): Solve $7p - 2 \leq 3p + 10$.

Subtract $3p$ from both sides to gather variables on the left: $4p - 2 \leq 10$.
Add 2 to both sides to gather constants on the right: $4p \leq 12$.
Divide by 4 to get $p \leq 3$.

• Example 3 (Negative division): Solve $5(k - 2) > -20$.

Distribute to get $5k - 10 > -20$.
Add 10 to both sides: $5k > -10$.
Divide by 5 to get $k > -2$. (The sign stays the same because we divided by a positive 5).

Explanation

Solving a multi-step inequality uses the exact same strategy as solving a multi-step equation: clean up both sides, move the letters to one team and the numbers to the other, and then isolate the variable. The only difference is the golden rule of inequalities—you must stay highly alert during the very last step. If you divide or multiply by a negative number to get the variable by itself, you must flip the inequality symbol.

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.