Algebra Review

Reading

• Review the algebra cheatsheet at the front of the book.
• Read sections 1.1, 1.2

Practice problems

• Section 1.1: 11, 15, 17, 23
• Section 1.2: 11, 13, 15, 22, 25, 35, 37

Notes

Algebra

• Multiplication distributes over addition
• Identities for \((x\pm y)^2\), \(x^2-y^2\), \(x^3\pm y^3\)
  • Question: Simplify \((x-y)^2 + (x+y)^2\)
  • Conclusion: If a integer is the sum of two squares, then so is its double
  • What about \((x+y)^2 - (x-y)^2\)?
  • Conclusion: The product of two even numbers is a difference of squares
• Multiplicative inverses
• Division is multiplication by the inverse
  • \(\frac{a}{b} = a\frac{1}{b}\)
  • \(\frac{c}{b/a} = \frac{ac}{b}\)
• Solving a linear equation such as \(5x+3 = 6x - 1\)
• Find the error in proof that \(0 = 1\)
• Solving an equation such as \((x-2)(x+1)(x-3)=0\)
• Linear functions and linear equations
  • Rise-over-run interpretation
  • Slope-intersect, point-slope, point-point forms
  • Does this describe all lines in the plane?
• Solving a quadratic equation such as \(2x^2 - 6x + 4 = 0\)
  • Quadratic formula and completing the square
  • Graph of quadratic

Inequalities

• Trichotomy:
  • For any two numbers \(a, b\), either \(a<b\) or \(a=b\) or \(a>b\)
  • Every number is either positive, zero, or negative
• Inequalities and addition
• Inequalities and multiplication (when does the direction reverse?)
• Inequalities comparing 0 to a product, like \((x-2)(x+3) > 0\).
  • What if we had three or four factors?
• Absolute value
• Triangle inequality
• Various interpretations of the inequality \(|x-a| < r\)
• Question: If \(|x-1| < 2\), then:
  • Is it true that \(|x-1| < 1\)?
  • Is it true that \(|x-1| < 3\)?
  • Is it true that \(x > -3\)?
  • Is it true that \(x > 0\)?