#Linear Equations and Inequalities: Your Night-Before-the-Test Guide

Hey there! Let's get you feeling super confident about linear equations and inequalities. Think of this as your ultimate cheat sheet – everything you need, nothing you don't. Let's dive in!

#Linear Equations: The Basics

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• Form: $$ax + b = c$$, where 'a', 'b', and 'c' are real numbers and $$a ≠ 0$$. It's all about that straight line!
• Goal: Isolate the variable (usually 'x') on one side of the equation.
• Solution: The value of 'x' that makes the equation true.

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• "Undo" Operations: Use opposite operations to isolate 'x'.
  • If it's addition, subtract. If it's multiplication, divide.
  • Remember: Whatever you do to one side, you must do to the other!
• Simplify: Combine like terms and use the distributive property (if needed) to clean things up.
  • Distributive property: $$a(b+c) = ab + ac$$

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• One Solution: A single value of 'x' works. Example: $$2x + 3 = 7$$ (x = 2)

• No Solution (Inconsistent): You end up with a false statement, like $$3 = 5$$. Example: $$2x + 5 = 2x + 8$$.

• Infinitely Many Solutions (Identity): The equation is true for any value of 'x'. Example: $$2x + 4 = 2(x + 2)$$.

• Verification: Always plug your solution back into the original equation to check your work.