Equations with Rational Expressions

1
Equations with Rational Expressions

• Section 7.6

2
Remember

• To SOLVE an equation means to find the value or
  values of the variable(s) that make(s) the
  equation a true statement.
• Thus far weve solved LINEAR (first degree)
  equations in one variable (Section 2.12.2).
• Weve solved QUADRATIC (second degree equations)
  by factoring (a different method than that used
  for solving first degree equations).
• Weve GRAPHED functions these are pictures of
  the infinite number of solutions of equations in
  TWO variables.

3
Remember (contd)

• We are now studying/exploring the algebra of
  rational expressions/equations/functions.
• Thus far we have simplified rational expressions.
  We have reduced an expression and performed the
  operations of addition, subtraction,
  multiplication, and division.
• We have also examined the graphs of rational
  functions. Remember, a function has an input
  value and an output value an x and y.

4
Solving Rational Equations

• We now turn our attention to solving rational
  equations.
• By definition, those are equations that contain
  at least one rational expression.
• We have the knowledge already to solve these
  equations.
• Remember, we can add, subtract, multiply and
  divide both sides of any equation by any nonzero
  constant. This produces an equivalent equation.
• With rational expressions, we simply have to make
  sure that an excluded value is NOT a solution.

5
Examples
6
Procedures

• Find the restricted values (excluded values)
  FIRST so that they are not inadvertently made a
  part of the solution.
• Remove any grouping distributive prop.
• Multiply through the entire equation (each term
  on both sides) by the LCD to clear fractions.
• Solve the resulting equation (identify it as
  linear/quadratic/or higher degree and solve
  appropriately).
• Reject any solutions that are excluded values
  they CANNOT be solutions.

7
Looking at the Big Picture
8
Reminders

• CHECK solutions (analytically, using the STOre
  feature, using the TABLE feature, or
  graphically).
• Do not enclose multiple solutions to these
  equations in parentheses! Recall, parentheses
  are a type of notation that indicates either an
  ordered pair of numbers or an interval of
  numbers.
• You can only clear fractions by multiplying
  through by an LCD in an EQUATION. NEVER do this
  to an EXPRESSION ?