Solving Inequalities in one Variable

1
Solving Inequalities in one Variable

• 2.9

2
Solving Polynomial Inequalities

• Put all terms on the left side and zero on the
  other. (Call this inequality 2.)
• Simplify the terms and then factor.
• For a cubic and higher, use the rational zeros
  theorem, the calculator, and synthetic division
  to help factor is needed.
• For a quadratic, factor by hand, or with the
  quadratic formula.
• For a linear, solve for x.

3
Solving Polynomial Inequalities

• Draw a number line.
• Plot each zero on the number line.
• The zeros separate the number line into
  intervals. Choose a number that lies in each
  interval and plug it in to inequality two.

4
Solving Polynomial Inequalities

• Write plus signs in the intervals where the
  chosen number yields a positive result and vice
  versa.

5
Solving Polynomial Inequalities

• Choose the negative intervals if the inequality
  is a less than. Choose the positive intervals if
  the inequality is a greater than. Include the
  zeroes if the less than or greater than includes
  an equal sign.
• Write your answers in interval notation.

6
Solving Polynomial Inequalities Graphically

• Put all the terms on one side with zero on the
  other.
• Put the result in y1.
• Choose the intervals where the graph is below the
  x-axis if the inequality is a less than. Choose
  the intervals where the graph is above the x-axis
  if the inequality is a greater than. Include the
  zeroes if the less than or greater than includes
  an equal sign.
• Write your answers in interval notation.

7
Solving Rational Inequalities Algebraically

• Rearrange the terms so that 0 is on the right
  side.
• Get a single fraction on the left, by creating a
  common denominator and adding the algebraic
  fractions.
• Find the zeros of f by setting the numerator
  equal to zero.
• Find the values at which f is undefined by
  setting the denominator equal to zero. (or
  negatives under a square root)

8
Solving Rational Inequalities Algebraically

• Draw a number line and plot the zeros and the
  values that make the function undefined.
• Select a number in each interval and plug the
  number into the original function with zero on
  one side. If the result is positive, then put
  pluses in that section, otherwise put minuses.

9
Solving Rational Inequalities Algebraically

• Choose the negative intervals if the inequality
  is a less than. Choose the positive intervals if
  the inequality is a greater than. Include the
  zeros(not the undefined) if the less than or
  greater than includes an equal sign.

10
Solving Rational Inequalities Graphically

• Put the left side of the inequality in y1 making
  sure that the entire numerator and denominator
  are surrounded by parenthesis in a fraction.
• Put the right side in y2.
• Use 2nd Calc Intersect to find where the graphs
  intersect.
• Plot these points on a number line.

11
Solving Rational Inequalities Graphically

• Place pluses in the sections where the graph on
  the left is lt, gt,, or (depending on the sign
  in the original problem) the graph on the right.
• Write these sections in interval notation.