Overview
This lecture covers key Algebra 1 problem types and strategies for solving equations, inequalities, word problems, absolute value, radicals, rational equations, and functions.
Solving One-Step and Two-Step Equations
- To solve x + 2 = 5, subtract 2 from both sides: x = 3.
- For 2x + 3 = 11, subtract 3 from both sides (2x = 8), then divide by 2: x = 4.
- Always use the opposite operation to isolate x: subtract for addition, divide for multiplication.
Multi-Step Equations & Order of Operations
- For 3x² + 8 = 20, subtract 8 (3x² = 12), divide by 3 (x² = 4), then square root: x = 2.
- Use the reverse order of operations: handle addition/subtraction before multiplication/division, and exponents last.
Equations with Variables on Both Sides
- For 4x + 5 = 9 + 2x, subtract 2x from both sides (2x + 5 = 9), then solve as a two-step equation for x = 2.
Absolute Value Equations
- For |x + 3| = 7, set x + 3 = 7 and x + 3 = -7; solve both for x = 4 and x = -10.
- For |x + 1| + 6 = 9, subtract 6 (|x + 1| = 3), then x + 1 = 3 and x + 1 = -3; x = 2 or x = -4.
Radical and Rational Equations
- For √(x + 3) - 2 = 1, add 2 (√(x + 3) = 3), square both sides (x + 3 = 9), so x = 6.
- For 4/(x-5) = 3/x, cross-multiply (4x = 3x - 15), subtract 3x (x = -15).
Formula Rearrangement
- To solve y = mx + b for x, subtract b (y - b = mx), divide by m: x = (y - b)/m.
Inequalities
- For -3x + 1 > 7, subtract 1 (-3x > 6), divide by -3 (reverse sign): x < -2.
- For -3 < x + 8 < 20, subtract 8 throughout: -11 < x < 12.
Graphing Inequalities
- For x > -4, draw an open circle at -4 on the number line, arrow to the right.
Word Problems (Two-Step Equations)
- For 2,500 gallons in 20 boxes, 100 gallons left: 20x + 100 = 2,500; solve to find x = 120 gallons per box.
- For "five added to thrice Michael’s age is 50": 5 + 3x = 50; solve to get x = 15 years.
Functions and Relations
- A function assigns each input (x) to exactly one output (y); an input with two outputs is not a function.
- Example: input 3 with outputs 6 and 8 is not a function.
Key Terms & Definitions
- Order of Operations — The rule for which mathematical operations to perform first (PEMDAS).
- Absolute Value — The distance of a number from zero, always positive.
- Radical Equation — An equation where the variable is inside a root.
- Rational Equation — An equation involving variables in denominators.
- Function — A relation where each input has exactly one output.
Action Items / Next Steps
- Practice additional problems on solving equations and inequalities.
- Review order of operations and function definitions.
- Watch more example videos for reinforcement.