Comprehensive Algebra Notes

Solving Linear Equations and Graphing Inequalities

Linear equations might look intimidating, but they follow a simple pattern once you get the hang of it. The key is working systematically through each step without rushing.

Let's break down the equation 3$2x + 4$ - 5 = 3. First, expand the brackets by multiplying everything inside by 3, giving you 6x + 12 - 5 = 3. Then simplify the left side to get 6x + 7 = 3. Now isolate the x term by subtracting 7 from both sides: 6x = -4. Finally, divide both sides by 6 to get x = -2/3.

Inequalities work similarly to equations, but instead of an equals sign, you're dealing with less than (≤) or greater than (≥) symbols. The inequality x ≤ 2 - 8 simplifies to x ≤ -6, meaning any number less than or equal to -6 satisfies this condition.

Quick Tip: When graphing inequalities, use a filled circle for ≤ or ≥ (the number is included) and an open circle for < or > (the number isn't included). The different number sets (ℝ for real numbers, ℤ for integers, ℕ for natural numbers) will change which values you can actually plot on your number line.