Foundations of Solving Equations

TL;DR

Solving equations means finding the value that makes both sides equal by isolating the variable. You achieve this by performing inverse operations to "undo" what's been done to the variable. Remember, whatever you do to one side of the equation, you must do to the other to keep it balanced.

1. The Mental Model

Think of an equation like a balanced scale. To keep it balanced, any operation you perform on one side must also be performed on the other side. Your goal is to get the "mystery item" (the variable) by itself on one side.

2. The Core Material

When you're solving an equation, you're essentially "unwinding" the operations that were performed on the variable. This means using inverse operations.

Inverse Operations

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• Addition is the inverse of Subtraction.
• Multiplication is the inverse of Division.

Solving Two-Step Equations

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A two-step equation typically involves one addition/subtraction and one multiplication/division. The key is to undo these operations in reverse order of how you'd normally solve them (think PEMDAS in reverse).

1. Undo addition or subtraction first.
2. Undo multiplication or division second.

Let's look at the process:

graph TD
    A["Start with the equation"] --> B["Identify the variable (e.g., x)"]
    B --> C["Look at the operations around the variable"]
    C --> D{"Is there addition or subtraction?"}
    D -- Yes --> E["Perform the inverse operation (subtract to undo addition, add to undo subtraction) to BOTH sides"]
    D -- No --> F{"Is there multiplication or division?"}
    E --> F
    F -- Yes --> G["Perform the inverse operation (divide to undo multiplication, multiply to undo division) to BOTH sides"]
    F -- No --> H["Variable is isolated!"]
    G --> H
    H --> I["Check your answer by plugging it back into the original equation"]
    I --> J["End"]

Solving Multi-Step Equations

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Multi-step equations involve more than two operations. They often require you to simplify expressions on one or both sides before you start using inverse operations.

1. Distribute any numbers outside parentheses.
2. Combine like terms on each side of the equation.
3. Move all variable terms to one side of the equation (usually the left).
4. Move all constant terms to the other side of the equation (usually the right).
5. Now you have a two-step equation, so solve it using inverse operations (undo addition/subtraction, then undo multiplication/division).

Foundations of Inequalities

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Solving inequalities is very similar to solving equations, with one crucial difference:

• When you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign.

For example:
If -2x < 10, you divide by -2. So, x > -5 (the < flips to >).

The steps for two-step and multi-step inequalities are the same as for equations, but remember that sign flip rule!

3. Worked Example

Let's solve the multi-step equation: 3(x - 2) + 5x = 10

1. Distribute: 3x - 6 + 5x = 10
2. Combine like terms: (3x + 5x) - 6 = 10 which simplifies to 8x - 6 = 10
3. Undo subtraction: Add 6 to both sides.
   8x - 6 + 6 = 10 + 6
8x = 16
4. Undo multiplication: Divide both sides by 8.
   8x / 8 = 16 / 8
x = 2

Check: 3(2 - 2) + 5(2) = 3(0) + 10 = 0 + 10 = 10. The solution is correct!

4. Key Takeaways

• Your main goal is to isolate the variable by itself on one side of the equation or inequality.
• Always use inverse operations to "undo" what's been done to the variable.
• Whatever operation you perform on one side, you must perform on the other to maintain balance.
• When solving multi-step problems, simplify each side by distributing and combining like terms first.
• Remember the special rule for inequalities: flip the sign if you multiply or divide by a negative number.
• Always check your solution by plugging it back into the original problem.
• Work methodically, one step at a time, to avoid errors.

5. Now Try It

Spend 15 minutes practicing solving a mix of two-step equations, multi-step equations, and two-step inequalities. For each problem, write down every step you take, just like in the worked example, and then check your answer. You'll know you're successful if you consistently arrive at the correct variable value or inequality range by carefully applying inverse operations and remembering the inequality sign flip rule.