© William James Calhoun, : Solving Open Sentences Involving Absolute Value OBJECTIVE: You will be able to solve open sentences involving absolute value and graph the solutions. We need to start with a discussion of what absolute value means. Absolute value is a means of determining the distance from zero. If I ask for the absolute value of 4, the answer is “4” since 4 is four units away from zero. If I ask for the absolute value of -4, the answer is “4” since -4 is four units away from zero. |4| = 4|-4| = 4

© William James Calhoun, : Solving Open Sentences Involving Absolute Value units In non-graphical terms, |x - 0| = 4 tells us: x = 4 or x = -4 Now examine absolute values with variables in them. |x| = 4 can be rewritten as: |x - 0| = 4 Rewriting the absolute value this way helps to explain what is going on. The absolute value tells us, “The distance between x and zero is four units.” So, x if four units away from zero in either direction. Graphically:

© William James Calhoun, : Solving Open Sentences Involving Absolute Value EXAMPLE 1: Solve |x - 3| = 5. Use the definition of absolute value. This problem is saying, “The distance between x and 3 is 5 units.” Graphically: units Center on the 3. Run both directions 5 units. The solution is then: x = -2 or x = 8 {-2, 8} Another way to solve this problem without graphing it first follows. The problem tells us that the distance between x and 3 is 5 units, so we know either: x - 3 = 5x - 3 = -5or solve these equations +3 x = 8x = -2 The solution is then: x = -2 or x = 8 {-2, 8} This second method is the CPM and is how I recommend you work these problems.

© William James Calhoun, : Solving Open Sentences Involving Absolute Value Now, what if we introduce inequalities into the absolute value mix. What does |x| < n mean? Again, this can be rewritten as: |x - 0| < n This tells us that, “The distance between x and zero is less than n.” Using a real number for n, we can see graphically what is going on. |x| < 4 aka |x - 0| < 4 Center on the 0. Run four units to the right, but do not include the number four since it is not “or equal to.” Now, run four units to the left, but do not include the number negative four since it is not “or equal to.” units The answer to the inequality, |x| < 4 is then: {x | -4 < x < 4} If |x| < 4 then we can see that either: x -4 We will use this to solve inequalities. Since the problem says the distance is “less than 4,” we need to shade everything between the center line and 4. Since the problem says the distance is “less than 4,” we need to shade everything between the center line and -4.

© William James Calhoun, : Solving Open Sentences Involving Absolute Value EXAMPLE 2: Solve |3 + 2x| < 11 and graph the solution set. Use what we just saw to rewrite the problem as two inequalities connected with “and.” 3 + 2x < x > -11and Notice the inequality switch and sign change on the 11! Now solve the inequalities. -3 2x < 82x > x < 4x > -7and {x | -7 < x < 4} Graph it:

© William James Calhoun, : Solving Open Sentences Involving Absolute Value When the problem was an absolute value “less than” something, like this: |x - #| < # the solution had “and” in it because the set of answers were contained between two numbers. When the problem is an absolute value “greater than” something, like this: |x - #| > # the solution will be different. The solution set is not contained by the end number. The solution set will be outside the bounds of the end number. The solution will NOT contain “and”, so it must contain… OR.

© William James Calhoun, : Solving Open Sentences Involving Absolute Value Look at |x| > 4. Remember this can be written as: |x - 0| > 4. This tells us, “The distance between x and zero is greater than 4 units.” Graphically: Center on the 0. Run four units to the right, but do not include the number four since it is not “or equal to.” Now, run four units to the left, but do not include the number negative four since it is not “or equal to.” units The solution set is: {x | x 4} Out of set-builder notation: x > 4 or x < -4 We will solve greater than absolute value inequalities the same way as lesser than absolute value inequalities. Just remember: < yields an and answer, and > yields an or answer. Since the problem says the distance is “greater than 4,” we need to shade everything to the right of 4. Since the problem says the distance is “greater than 4,” we need to shade everything to the left of -4.

© William James Calhoun, : Solving Open Sentences Involving Absolute Value EXAMPLE 3: Solve |5 + 2y|  3 and graph the solution set Use what we just saw to rewrite the problem as two inequalities connected with “or.” 5 + 2y  y  -3 or Notice the inequality switch and sign change on the 11! Now solve the inequalities. -5 2y  -22y  -8 2 y  -1y  -4 or {y | y  -1 or y  -4} Graph it:

© William James Calhoun, : Solving Open Sentences Involving Absolute Value HOMEWORK Page 424 # odd