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Absolute Value Equations and Inequalities

2.5. Absolute Value Equations and Inequalities. Evaluate and graph the absolute value function Solve absolute value equations Solve absolute value inequalities. Definition of the Absolute Value Function. The graph of y = | x |. V-shaped Cannot be represented by single linear function.

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Absolute Value Equations and Inequalities

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  1. 2.5 Absolute Value Equations and Inequalities Evaluate and graph the absolute value function Solve absolute value equations Solve absolute value inequalities

  2. Definition of the Absolute Value Function The graph of y = |x| V-shaped Cannot be represented by single linear function. (piece-wise linear)

  3. Alternate Definition of the Absolute Value Function That is, regardless of whether a real number x is positive or negative, the expression equals the absolute value of x. Examples:

  4. Example (Analyzing the graph of y = |mx + b|) For the following linear function f, graph y = f (x) and y = |f (x)| separately. Discuss how the absolute value affects the graph of f. f(x) = –2x + 4 The graph of y = |–2x + 4| is a reflection of f across the x-axis when y = –2x + 4 is below the x-axis.

  5. Intersecting Functions We need to calculate where these two functions intersect. The following graph shows us that they intersect where they are equal . We can do this symbolically by setting them equal to each other and solving for x.

  6. Intersecting Functions Now lets solve the same problem using the TI. http://calculator.maconstate.edu/points_intersection/index.html

  7. Example : Solving an equation by absolute value equation by intersection Graph Y1 = abs(2X + 5) and Y2 = 2 Solve the equation |2x + 5| = 2 graphically Solution: x = –1.5 , y = 2 Solution: x = –3.5, y = 2 Solve the equation |2x + 5| = 2 using a table Table Solution Solutions to y1 = y2 are –3.5 and –1.5.

  8. Example : Solve the equation |2x + 5| = 2 symbolically Remove the absolute value bars by using Plus 2 case Minus 2 case

  9. Absolute Value Inequalities Solution to |ax + b| <k is in green. Solution to |ax + b| >k is in green.

  10. Solving Absolute Value Inequalities 1. |ax + b| < k is equivalent to s1 < x < s2 2. |ax + b| > k is equivalent to x < s1orx> s2 Solve using “=“ Let solutions to |ax + b| =k be s1 and s2, where s1 < s2 and k > 0. Solve the inequality |2x – 5| ≤ 6 symbolically Solution set: Solve the inequality |5 - x| > 3 symbolically Solution set:

  11. Solving Absolute Value Inequalities using the Alternative Method Solve using “<>“ 1. |ax + b| <k is equivalent to –k < ax + b< k. 2. |ax + b| >k is equivalent to ax + b < –k orax + b > –k Solve the inequality |4 – 5x | ≤ 3 < means put in between!!! In interval notation, solution is > means use OR!!! Solve the inequality |-4x – 6 | > 2 In interval notation, solution is

  12. Key Ideas for this section: • What is the absolute value function? • How do we solve absolute value equations by intersection? • How do we solve absolute value equations by using a table? • How do we solve inequalities involving absolute values symbolically? • How do we solve inequalities using the alternative method?

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