Overview
This lecture covers practice questions on geometry, algebraic inequalities, number theory, and equation solving, focusing on identifying counterexamples, solving inequalities, and verifying solutions.
Geometry Counterexample (Question 91)
- If side lengths of a square are doubled, the area does not always increase by at least four square units.
- Example: Increasing side from 1 to 2, area goes from 1 to 4 (increase of 3), disproving the claim.
Absolute Value Inequalities (Question 92)
- For |x - 5| > 0, the solution splits into x - 5 > 0 and x - 5 < 0.
- This simplifies to x > 5 or x < 5; x cannot equal 5.
- Plugging in values confirms x = 5 does not satisfy the inequality.
Number Theory—Factors (Question 93)
- If a and b are positive integers, and a is a factor of b, then a ≤ b.
- A factor divides exactly into a number; e.g., 5 is a factor of 35.
- The statement "a > b" must be false since a cannot be greater than b.
- The following are always true: b = a × k (for some integer k), b is a multiple of a, and b/a is an integer.
Equation Solutions (Question 94)
- To find which value is not a solution, substitute each choice into the equation.
- For x³ - 9x = 16x:
- x = 0 → 0 = 0 (solution)
- x = 1 → -8 ≠16 (not a solution)
- x = 5 → 80 = 80 (solution)
- Factoring x³ - 9x - 16x = 0 leads to x(x² - 25) = 0, so solutions are x = 0, x = 5, x = -5.
Key Terms & Definitions
- Counterexample — A specific case which disproves a general statement.
- Absolute Value — The distance of a number from zero; written |x|.
- Factor — A number that divides another number without leaving a remainder.
- Solution to an Equation — A value that makes the equation true when substituted.
Action Items / Next Steps
- Review factoring techniques for cubic equations.
- Practice solving absolute value inequalities and checking solutions.
- Complete additional practice problems on identifying counterexamples in geometry and number theory.