Solving Linear Inequalities

TL;DR

Solving linear inequalities is very similar to solving linear equations, but you'll get a range of solutions instead of a single number. The key difference is remembering to flip the inequality sign if you multiply or divide by a negative number. Graphing these solutions on a number line helps you visualize all possible answers.

1. The Mental Model

Think of an inequality like a seesaw that isn't perfectly balanced. Instead of finding the exact weight that makes it level (an equation), you're finding all the weights that keep it tilted in a certain direction.

2. The Core Material

Solving linear inequalities means finding the set of all numbers that make the inequality true. The steps are almost identical to solving equations, with one crucial difference.

The Basic Steps

1. Simplify Both Sides: Combine like terms and distribute if necessary, just like with equations.
2. Isolate the Variable Term: Use addition or subtraction to get all terms with the variable on one side and constant terms on the other.
3. Isolate the Variable: Use multiplication or division to get the variable by itself.

The BIG Rule Change

When you multiply or divide both sides of an inequality by a negative number, you MUST flip the direction of the inequality sign.

• > becomes <
• < becomes >
• ≥ becomes ≤
• ≤ becomes ≥

If you multiply or divide by a positive number, or if you add/subtract any number, the sign stays the same.

Graphing Solutions

Since inequalities have many solutions, we often graph them on a number line:

• Open Circle (or Parenthesis): Used for > or <. This means the number itself is NOT included in the solution.
• Closed Circle (or Bracket): Used for ≥ or ≤. This means the number itself IS included in the solution.
• Arrow: Points in the direction of the solutions.

graph TD
    Start["Begin with the inequality"] --> Simplify["Simplify both sides (combine like terms, distribute)"]
    Simplify --> VariableTerm["Add/Subtract to get variable terms on one side"]
    VariableTerm --> ConstantTerm["Add/Subtract to get constant terms on the other side"]
    ConstantTerm --> MultiplyDivide["Multiply/Divide to isolate the variable"]
    MultiplyDivide --> CheckNegative["Did you multiply/divide by a negative number?"]
    CheckNegative -- Yes --> FlipSign["FLIP the inequality sign!"]
    CheckNegative -- No --> KeepSign["Keep the inequality sign"]
    FlipSign --> Graph["Graph the solution on a number line"]
    KeepSign --> Graph
    Graph --> End["End"]

Examples of Graphing:

• x > 3: Open circle at 3, arrow pointing right.
• x ≤ -1: Closed circle at -1, arrow pointing left.

3. Worked Example

Let's solve and graph the inequality: -2x + 5 < 15

1. Simplify: No simplification needed on either side.

2. Isolate the variable term: Subtract 5 from both sides.
   -2x + 5 - 5 < 15 - 5
-2x < 10

3. Isolate the variable: Divide both sides by -2.

   • Crucial Step: We're dividing by a negative number, so we MUST flip the inequality sign.
     -2x / -2 > 10 / -2 (Notice the < flipped to >)
     x > -5

So, the solution is all numbers greater than -5.

Graphing:
On a number line, you would put an open circle at -5 (because x must be greater than -5, not equal to it). Then, draw an arrow extending to the right from -5, indicating that all numbers in that direction are solutions.

4. Key Takeaways

• Solving inequalities uses most of the same steps as solving equations.
• The most important rule: Flip the inequality sign when multiplying or dividing both sides by a negative number.
• Adding or subtracting any number (positive or negative) does not change the inequality sign.
• Graph solutions on a number line using an open circle for > or < and a closed circle for ≥ or ≤.
• An open circle means the boundary number is not part of the solution; a closed circle means it is.
• The arrow on the number line shows the direction of all possible solutions.

Common Mistakes to Avoid:
- Forgetting to flip the sign when dividing or multiplying by a negative number. This is the #1 mistake!
- Confusing open vs. closed circles when graphing.
- Getting the direction of the arrow wrong on the number line.
- Trying to flip the sign when adding or subtracting—only do it for multiplication/division by negatives.

5. Now Try It

Solve the following inequality and then graph its solution on a number line:

4 - 3x ≥ 19

What to do: Follow the steps: simplify, isolate the variable term, isolate the variable (remembering the sign flip rule!), and then draw a simple number line graph.

What success looks like: Your final solution will be in the form x [inequality sign] [number], and your graph will correctly show either an open or closed circle at the right number, with the arrow pointing in the correct direction.