1. Absolute Value Inequalities Algebra

2. Farmer Bob has four sheep. One day, he notices that they are standing in such a way that they are all the same distance away from each other. That is to say, the distance between any two of the four sheep is the same. How can this be so? From Lateral Thinking Puzzlers  1991 by Paul Sloane

3. Solving an Absolute-Value Inequalities  8  7  6  5  4  3  2  1 0 1 2 3 4 5 6 7 8

4. Graphing Absolute Value When an absolute value is greater than the variable you have a disjunction to graph. When an absolute value is less than the variable you have a conjunction to graph.

5. This can be written as 1  x  7. Solve | x  4 | < 3 Solving an Absolute-Value Inequality x  4 IS POSITIVE x  4 IS NEGATIVE | x  4 |  3 x  4   3 x  7 | x  4 |  3 x  4   3 x  1 Reverse inequality symbol.  The solution is all real numbers greater than 1 and less than 7.

6. 2x  1   9 | 2x  1 |  3  6 | 2x  1 |  9 2x   10 2x + 1 IS NEGATIVE  x   5 Solve | 2x  1 |  3  6 and graph the solution. | 2x  1 |  3  6 | 2x  1 |  9 2x  1  +9 2x  8 2x + 1 IS POSITIVE x  4 Solving an Absolute-Value Inequality Reverse inequality symbol. | 2x  1 |  3  6 | 2x  1 |  9 2x  1  +9 x  4 2x  8 | 2x  1 |  3  6 | 2x  1 |  9 2x  1   9 2x   10 x   5 2x + 1 IS POSITIVE 2x + 1 IS NEGATIVE  6  5  4  3  2  1 0 1 2 3 4 5 6 The solution is all real numbers greater than or equal to 4 or less than or equal to  5. This can be written as the compound inequality x   5 or x  4.  5 5 4.4.

7. Strange Results True for All Real Numbers, since absolute value is always positive, and therefore greater than any negative. No Solution Ø. Positive numbers are never less than negative numbers.

8. Examples or Check and verify on a number line. Numbers above 6 or below -1 keep the absolute value greater than 7. Numbers between them make the absolute value less than 7.

9. Key Skills Solve absolute-value inequalities. Solve | x – 4 |  5. x – 4 is positive x – 4  5 x – 4 is negative x  9 Case 1: Case 2: x – 4  –5 x  –1 solution: –1  x  9

10. Key Skills Solve absolute-value inequalities. Solve |4 x – 2 |  -18. Exception alert!!!! When the absolute value equals a negative value, there is no solution.

11. Key Skills Solve absolute-value inequalities. Solve |2 x – 6 |  18. 2x – 6 is positive 2x – 6  18 2x – 6 is negative x  12 Case 1: Case 2: 2x - 6  –18 x  –6 Solution: –6  x  12 2x  24 2x  –12 TRY THIS

12. Key Skills Solve absolute-value inequalities. Solve |3 x – 2 |  -4. Exception alert!!!! When the absolute value equals a negative value, there is no solution. TRY THIS