Solving Multi-Step Inequalities

TL;DR

Solving multi-step inequalities is very similar to solving multi-step equations, but with one crucial difference: you flip the inequality sign when multiplying or dividing by a negative number. Your goal is to isolate the variable, just like with equations. The solution set will usually include many numbers, which you can graph on a number line.

1. The Mental Model

Think of an inequality like a seesaw that might tip one way or the other, or stay balanced. Whatever you do to one side, you must do to the other to keep the relationship true. If you do something that fundamentally changes "less than" to "greater than" (like multiplying by a negative), you need to flip the seesaw.

2. The Core Material

Solving multi-step inequalities combines what you've learned about two-step equations with the special rule for inequalities. The steps are generally:
1. Simplify both sides of the inequality. This means combining like terms, and distributing if necessary.
2. Move variable terms to one side and constant terms to the other, using addition or subtraction.
3. Isolate the variable using multiplication or division.

The Golden Rule of Inequalities

Photo by Jan van der Wolf on Pexels

This is the most important part:
* If you multiply or divide both sides of an inequality by a negative number, you MUST flip the inequality sign.

Let's look at an example:
2x + 5 > 15
2x > 10 (subtract 5 from both sides)
x > 5 (divide by positive 2, sign stays the same)

Now, with a negative:
-2x + 5 > 15
-2x > 10 (subtract 5 from both sides)
x < -5 (divide by negative 2, flip the sign!)

Graphing the Solution

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After you solve an inequality, it's often helpful to graph the solution on a number line.
* Use an open circle for > or < (the number isn't included).
* Use a closed circle for ≥ or ≤ (the number is included).
* Draw an arrow in the direction that satisfies the inequality.

graph TD
    A["Start with the inequality"] --> B{"Simplify both sides (distribute, combine like terms)"};
    B --> C{"Add or subtract to move variable terms to one side"};
    C --> D{"Add or subtract to move constant terms to the other side"};
    D --> E{"Multiply or divide to isolate the variable"};
    E --> F{"Was multiplication or division by a NEGATIVE number used in step E?"};
    F -- Yes --> G["Flip the inequality sign"];
    F -- No --> H["Keep the inequality sign the same"];
    G --> I["Write the solution"];
    H --> I;
    I --> J["Graph the solution on a number line"];

3. Worked Example

Let's solve and graph: 3(x - 2) - 5x ≤ 10

1. Distribute:
   3x - 6 - 5x ≤ 10
2. Combine like terms:
   -2x - 6 ≤ 10
3. Add 6 to both sides:
   -2x ≤ 16
4. Divide by -2 (and remember the golden rule!):
   x ≥ -8 (The sign flipped because we divided by a negative!)

Graphing: You'd draw a number line, put a closed circle at -8, and draw an arrow pointing to the right (towards positive infinity), indicating all numbers greater than or equal to -8 are solutions.

4. Key Takeaways

• Simplify each side of the inequality first by distributing and combining like terms.
• Move all terms with the variable to one side and constant terms to the other.
• The only time you flip the inequality sign is when you multiply or divide both sides by a negative number.
• An open circle on a number line means the endpoint is not included (<, >).
• A closed circle means the endpoint is included (≤, ≥).

Common Mistakes to Avoid:
- Forgetting to flip the inequality sign when multiplying or dividing by a negative number.
- Incorrectly combining like terms or distributing.
- Mixing up open and closed circles when graphing.
- Forgetting that variables on both sides need to be moved to consolidate them.

5. Now Try It

Solve the following inequality and then sketch its solution on a number line:
4 - 2(3y + 1) < 18

What to do: Follow the steps: distribute, combine, add/subtract, multiply/divide (remembering the sign flip rule).
What success looks like: You'll have an inequality with 'y' isolated and the correct inequality sign, along with a number line showing the correct circle type and direction.