1.8 Absolute Value Equations and Inequalities

1.8 Absolute Value Equations and Inequalities

• 1.
  Absolute Value Equations Chapter1 Equations and Inequalities
• 2.
  Concepts and Objectives Solve absolute value equations and inequalities
• 3.
  Absolute Value  Youshould recall that the absolute value of a number a, written |a|, gives the distance from a to 0 on a number line.  By this definition, the equation |x| = 3 can be solved by finding all real numbers at a distance of 3 units from 0. Both of the numbers 3 and ‒3 satisfy this equation, so the solution set is {‒3, 3}.
• 4.
  Absolute Value (cont.) The solution set for the equation must include both a and –a.  Example: Solve x a  9 4 7x
• 5.
  Absolute Value  Thesolution set for the equation must include both a and –a.  Example: Solve The solution set is x a  9 4 7x  9 4 7x  9 4 7x   4 2x  4 16x  1 2 x  4x or       1 ,4 2
• 6.
  Absolute Value  Forabsolute value inequalities, we make use of the following two properties:  |a| < b if and only if –b < a < b.  |a| > b if and only if a < –b or a > b.  Example: Solve   5 8 6 14x
• 7.
  Absolute Value  Example:Solve The solution set is   5 8 6 14x or  5 8 8x  5 8 8x  5 8 8x  8 13x  8 3x  13 8 x  3 8 x              3 13 , , 8 8
• 8.
  Special Cases  Sincean absolute value expression is always nonnegative:  Expressions such as |2 – 5x| > –4 are always true. Its solution set includes all real numbers, that is, –, .  Expressions such as |4x – 7| < –3 are always false— that is, it has no solution.  The absolute value of 0 is equal to 0, so you can solve it as a regular equation.
• 9.
  Classwork  College Algebra Page 163: 10-22 (evens), page 155: 26-36 (evens), page 144: 36-44 (4), 78