Solving Linear Equations and Inequalities

TL;DR

Linear equations have a single exact answer for the unknown variable, found by isolating it. Inequalities, however, have a range of answers, and flipping the inequality sign is crucial when multiplying or dividing by a negative number. The goal for both is to get the variable by itself on one side.

1. The Mental Model

Think of an equation as a balanced scale; whatever you do to one side, you must do to the other to keep it balanced. An inequality is like a scale that's tipped, and your goal is still to find out what range of weights would keep it tipped in the same direction.

2. The Core Material

When you're solving linear equations or inequalities, you're trying to figure out what value (or range of values) a variable represents. You do this by performing operations (addition, subtraction, multiplication, division) to both sides until the variable is alone.

Isolating the Variable in Equations

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The key is to perform inverse operations. If a number is added to the variable, subtract it from both sides. If it's multiplied, divide both sides. Always aim to get the variable term by itself first, then the variable itself.

Let's say you have an equation like 2x + 5 = 11.
1. Subtract 5 from both sides: 2x + 5 - 5 = 11 - 5, which simplifies to 2x = 6.
2. Divide by 2 on both sides: 2x / 2 = 6 / 2, which simplifies to x = 3.
So, the solution is x = 3.

Solving Linear Inequalities

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The process for inequalities is almost identical to equations, with one very important exception: if you multiply or divide both sides by a negative number, you must flip the direction of the inequality sign.

For example, if you have x + 3 < 7:
1. Subtract 3 from both sides: x + 3 - 3 < 7 - 3, which simplifies to x < 4.

Now consider -2x >= 10:
1. Divide by -2 on both sides: -2x / -2 >= 10 / -2.
2. Flip the inequality sign because you divided by a negative number: x <= -5.

It's helpful to visualize the steps involved in tackling these problems.

graph TD
    A["Start: Equation or Inequality"] --> B{"Isolate Variable Term?"}
    B -- "No" --> C["Add/Subtract terms (inverse ops) to both sides"]
    C --> B
    B -- "Yes" --> D{"Isolate Variable?"}
    D -- "No" --> E["Multiply/Divide by coefficient (inverse ops) to both sides"]
    E --> F{"Multiplied/Divided by a negative number?"}
    F -- "Yes" --> G["FLIP INEQUALITY SIGN"]
    F -- "No" --> H["Keep sign as is"]
    G --> I["Result: Solution (x = #) or Solution Set (x < #)"]
    H --> I
    D -- "Yes" --> I

3. Worked Example

Let's solve the inequality 3(y - 2) + 7 <= 16.

1. Distribute the 3: 3y - 6 + 7 <= 16
2. Combine like terms: 3y + 1 <= 16
3. Subtract 1 from both sides: 3y + 1 - 1 <= 16 - 1, which gives 3y <= 15
4. Divide by 3 on both sides: 3y / 3 <= 15 / 3. We are dividing by a positive number, so the sign doesn't change.
5. Simplify: y <= 5

The solution is y <= 5. This means any number less than or equal to 5 will make the original inequality true.

4. Key Takeaways

• Always perform the same operation on both sides of the equation or inequality to maintain balance.
• Use inverse operations to "undo" what's being done to the variable.
• When solving inequalities, remember to flip the inequality sign ONLY when you multiply or divide both sides by a negative number.
• Equations have a single, specific solution.
• Inequalities have a range of solutions that can be graphed on a number line.
• Combine like terms on each side before trying to move terms across the equals/inequality sign.

Common Mistakes to Avoid:
- Forgetting to flip the inequality sign when dividing/multiplying by a negative.
- Only performing an operation on one side of the equation/inequality.
- Making arithmetic errors, especially with negative numbers.
- Not distributing correctly when parentheses are present.

5. Now Try It

Solve the inequality -5(x + 4) - 2x > 28. Your goal is to find the range of values for x that makes this statement true. A successful solution will clearly show each step, including any necessary sign flips, and end with x isolated on one side with the correct inequality sign and value.