Overview
This lecture provides an overview of key Algebra 1 question types and straightforward solving techniques. Topics include solving equations (one-step, two-step, and multi-step), handling variables on both sides, absolute value equations, radical and rational equations, changing the subject of a formula, inequalities, graphing inequalities, word problems, and understanding functions and relations.
Solving One-Step Equations
- The main goal is to isolate x by moving all other terms to the opposite side using the opposite operation.
- Example: For x + 2 = 5, subtract 2 from both sides to get x = 3.
- Remember: The opposite of addition is subtraction, multiplication is division, and exponentiation is taking the root.
Two-Step and Multi-Step Equations
- For equations with more than one operation, reverse the order of operations (work from addition/subtraction up to multiplication/division, then exponents/roots).
- Example: For 2x + 3 = 11:
- Subtract 3 from both sides: 2x = 8.
- Divide both sides by 2: x = 4.
- For 3x² + 8 = 20:
- Subtract 8: 3x² = 12.
- Divide by 3: x² = 4.
- Take the square root: x = 2.
Equations with Variables on Both Sides
- Move all x terms to one side and constants to the other.
- Example: 4x + 5 = 9 + 2x
- Subtract 2x from both sides: 2x + 5 = 9.
- Subtract 5: 2x = 4.
- Divide by 2: x = 2.
- Use the reversal of the order of operations to decide which terms to move first.
Absolute Value Equations
- For equations like |x + 3| = 7:
- Set up two equations: x + 3 = 7 and x + 3 = -7.
- Solve both: x = 4 or x = -10.
- If there are terms outside the absolute value, isolate the absolute value first.
- Example: |x + 1| + 6 = 9
- Subtract 6: |x + 1| = 3.
- Set up two equations: x + 1 = 3 and x + 1 = -3.
- Solve: x = 2 or x = -4.
Radical Equations
- Radical equations have the variable inside a root.
- Isolate the radical first, then eliminate the root by raising both sides to the appropriate power.
- Example: √(x + 3) - 2 = 1
- Add 2: √(x + 3) = 3.
- Square both sides: x + 3 = 9.
- Subtract 3: x = 6.
Rational Equations
- Rational equations have variables in the denominator.
- Remove fractions by cross-multiplying or using the least common denominator.
- Example: 4/(x - 5) = 3/x
- Cross-multiply: 4x = 3(x - 5).
- Expand: 4x = 3x - 15.
- Subtract 3x: x = -15.
Changing Subject of a Formula
- To solve for a specific variable, isolate it using the opposite operations in reverse order.
- Example: y = mx + b, solve for x:
- Subtract b: y - b = mx.
- Divide by m: x = (y - b)/m.
- This process is called transposing or changing the subject of the formula.
Inequalities
- Solve inequalities like equations, but reverse the inequality sign when multiplying or dividing by a negative.
- Example: -3x + 1 > 7
- Subtract 1: -3x > 6.
- Divide by -3 (reverse the sign): x < -2.
- For combined inequalities, perform the same operation on all parts.
- Example: -3 < x + 8 < 20
- Subtract 8: -11 < x < 12.
Graphing Inequalities
- Draw a number line and locate the key value.
- Use an unshaded circle for < or >, and a shaded circle for ≤ or ≥.
- Draw an arrow in the direction indicated by the inequality.
- Example: For x > -4, place an unshaded circle at -4 and draw an arrow to the right.
Word Problems with Equations
- Identify the unknown and represent it with a variable.
- Translate the problem into an equation and solve step by step.
- Example: To find gallons per box when 2,500 gallons are shipped in 20 boxes with 100 gallons left over:
- Set up: 20x + 100 = 2,500.
- Subtract 100: 20x = 2,400.
- Divide by 20: x = 120 gallons per box.
- For age problems: "Five added to thrice Michael’s age is 50."
- Let x = Michael’s age.
- Set up: 5 + 3x = 50.
- Subtract 5: 3x = 45.
- Divide by 3: x = 15.
Functions and Relations
- A function assigns exactly one output to each input.
- If an input has more than one output, it is not a function.
- Example: If input 3 maps to both 6 and 8, the relation is not a function.
- Multiple inputs can have the same output and still be a function.
Key Terms & Definitions
- One-step equation: An equation solved with a single operation.
- Order of operations: The sequence for solving expressions (PEMDAS).
- Absolute value: The distance from zero; always positive.
- Radical equation: An equation with variables inside a root.
- Rational equation: An equation with variables in the denominator.
- Inequality: A statement comparing two expressions using <, >, ≤, or ≥.
- Function: A relation where each input has only one output.
Action Items / Next Steps
- Practice solving all types of equations and inequalities discussed.
- Review and memorize the order of operations.
- Complete assigned homework problems for each equation type.
- Watch additional recommended videos for more examples and deeper understanding.
- Focus on identifying key values and translating word problems into equations for efficient problem-solving.