Introduction to College Algebra

Basics of Exponents

• Multiplying Exponents: When multiplying common bases, add the exponents.
  Example:

  • ( x^2 \times x^5 = x^{2+5} = x^7 )

• Dividing Exponents: When dividing, subtract the exponents.
  Example:

  • ( x^5 \div x^2 = x^{5-2} = x^3 )

• Negative Exponents: Move the base from numerator to denominator or vice versa and change the sign.
  Example:

  • ( x^4 \div x^7 = x^{4-7} = x^{-3} = \frac{1}{x^3} )

• Raising Exponents: Multiply the exponents.
  Example:

  • ( (x^3)^4 = x^{3 \times 4} = x^{12} )

• Zero Exponent: Any base raised to the zero power equals one.
  Example:

  • ( x^0 = 1 )

Simplifying Expressions and Combining Like Terms

• Combine like terms by adding coefficients.
  Example:
  • ( 5x + 7x = 12x )
  • ( 3 - 4 = -1 )

Example Problems

1. Expression: ( 3x^2 + 6x + 8 + 9x^2 + 7x - 5 )

   • Combine like terms:
   • ( (3 + 9)x^2 = 12x^2 )
   • ( (6 + 7)x = 13x )
   • ( 8 - 5 = 3 )
   • Result: ( 12x^2 + 13x + 3 )

2. Distributing Negative Signs:

   • Example: ( 5x^2 - 3x + 7 - (4x^2 + 8x + 11) )
   • Result: ( 5x^2 - 4x^2 - 3x + 8x + 7 + 11 )
   • Final Answer: ( x^2 + 5x + 18 )

Multiplying Binomials (FOIL Method)

• FOIL:
  • First: Multiply the first terms.
  • Outside: Multiply the outside terms.
  • Inside: Multiply the inside terms.
  • Last: Multiply the last terms.

Example Problem

• ((3x - 5)(2x - 6))
  • Result: ( 6x^2 - 18x + 10x + 30 = 6x^2 - 8x + 30 )

Solving Linear Equations

• Use inverse operations to isolate the variable.

  • Example:
    ( x + 6 = 11 )
  • Solution: ( x = 5 )

• For multiplication:

  • Example:
    ( 4x = 8 )
  • Solution: ( x = 2 )

Solving Inequalities

• Treat inequalities like equations but pay attention to sign changes when multiplying/dividing by negative numbers.
• Example:
  ( 2x + 5 > 11 )
  • Result: ( x > 3 )

Absolute Value Expressions

• Absolute value makes any number positive.
  Example:
  ( |x| = x ) if ( x \geq 0 )
  ( |x| = -x ) if ( x < 0 )

Solving Absolute Value Equations

• Example:
  • ( |2x + 3| = 11 ) leads to two equations:
    • ( 2x + 3 = 11 ) and ( 2x + 3 = -11 )

Graphing Linear Equations

• Slope-Intercept Form: ( y = mx + b )
  • ( m ) is the slope, ( b ) is the y-intercept.

Example Problem

1. Graph ( y = 2x - 3 ):
   • Y-intercept is ( (0, -3) )
   • Slope 2 (rise/run):
   • Plot additional points based on slope.

Functions and Composite Functions

• Composite Functions: ( f(g(x)) )
• Example:
  • If ( f(x) = 3x + 5 ) and ( g(x) = x^2 - 4 ), find ( f(g(2)) )
• Evaluate functions by substituting values.

Inverse Functions

• To find an inverse, switch ( x ) and ( y ) in the function and solve for ( y ).
  • Example: For ( f(x) = 7x - 5 ):
    • Inverse is ( f^{-1}(x) = \frac{x + 5}{7} )

Conclusion

• Review all topics covered, practice problems, and utilize resources for additional assistance.