Solving Absolute Value Equations Solving Absolute Value Equations Absolute Values Solving Absolute Value Equations Not all functions are defined in a simple way by a neat formula. Sometimes we have to break up the domain and define the function differently on different pieces of the domain. Absolute Value Equations 1 10/2/2013

Absolute Value Equations Solving Absolute Value Equations General Procedure Form is: for constants a, b and k ≥ 0 By definition: | ax + b | = k either | ax + b | = ax + b = k OR | ax + b | = –(ax + b) = k Solving Absolute Value Equations The simplest way to solve equations with absolute values is to replace the absolute value terms with their equivalent. We appeal to the definition of absolute value. For any number a we define | a | as a , for a ≥ 0 | a | –a , for a < 0 Similarly if | a x + b | = k for some k ≥ 0, then a x + b = ±k. Note: The solutions should be substituted into the original equation to verify that they produce a true statement. (ax + b) = –k ax + b = k + – Thus = 10/2/2013 Absolute Value Equations Absolute Value Equations 10/2/2013

Absolute Value Equations Solving Absolute Value Equations Example 1 Solve –3x – 2= 5 –3x – 2 = 5 or –3x – 2 = –5 –3x = 5 + 2 or –3x = –5 + 2 x = –7/3 or x = 1 Solution set is: { –7/3, 1 } Solving Absolute Value Equations: Example 1 We replace the absolute value terms with their equivalent by appealing to the definition of absolute value. For any number a we define | a | as a , for a ≥ 0 | a | –a , for a < 0 Similarly if | a x + b | = k for some k ≥ 0, then a x + b = ±k. In Example 1, k = 5, so either -3x – 2 = 5 or -3x – 2 = -5. We solve both inequalities revealing that either x = -7/3 or x = 1 as the two solutions for the original equation. The solution set is thus { -7/3, 1 }. Note: The solutions should be substituted into the original equation to verify that they produce a true statement. = 10/2/2013 Absolute Value Equations Absolute Value Equations 10/2/2013

Absolute Value Equations Solving Absolute Value Equations Example 2 Solve 3x= –6 Note: 3x ≥ 0 for all x Hence, 3x= –6 is FALSE for every value of x Thus there is no solution and thus solution set is Solving Absolute Value Equations: Example 2 We apply the principles we have just discussed – along with common sense – in solving some absolute-value equations. In Example 2 we simply apply common sense by noting that | 3x | is always greater than or equal to 0. So | 3x | can never equal -6, or any other negative number ! Hence this equation has no solution and the solution set is empty, that is { } or ∅. O or { } Note: O ≠ { } WHY ? 10/2/2013 Absolute Value Equations Absolute Value Equations 10/2/2013

Absolute Value Equations Solving Absolute Value Equations Example 3 Solve 2x – 3 – 4 = 7 2x – 3 = 7 + 4 = 11 2x – 3 = 11 or –(2x – 3 ) = 11 2x – 3 = 11 or 2x – 3 = –11 2x = 14 or 2x = –8 x = 7 or x = –4 Solution set is { –4, 7 } Solving Absolute Value Equations: Example 3 We apply the principles we have just discussed – along with common sense – in solving some absolute-value equations. In Example 3, in order to apply the rule that | a x + b | = ±k for some constant k, we must transform the equation into this form by adding 4 to both sides to get | 2x – 3 | = 11 which in turn gives two equations: 2x – 3 = 11 and 2x – 3 = -11 Solving these gives solutions 7 and -4, respectively. So solution set is { -4, 7}. 10/2/2013 Absolute Value Equations Absolute Value Equations 10/2/2013

Absolute Value Equations Solving Absolute Value Equations Example 4 Solve 9 + x= 3 – 2x Removing one absolute value at a time: 9 + x = 3 – 2x and 9 + x= –(3 – 2x) Same 9 + x = 3 – 2x –(9 + x) = –(3 – 2x) Solving Absolute Value Equations: Example 4 Here we have an equation with two absolute values, one on each side of the equation. The strategy is to remove the absolute values one at a time. Recall that | a | is either a or –a , depending on whether a < 0 or a ≥ 0. We remove the absolute values on the right side first. This gives either 3 – 2x or –(3 – 2x) on the right, both equal to the absolute value on the left. Then we apply the definition of absolute value to remove the absolute value on the left from these two equations. In the first of the two we get either 9 + x = 3 – 2x or –(9 + x) = 3 – 2x and in the second equation we get either –(9 + x) = –(3 – 2x) or 9 + x = –(3 – 2x) and we note that of these four equations the left hand two are equivalent and the right hand two are equivalent, as shown in the illustration. In effect the equation with two absolute values is solved exactly like an equation with only one side in absolute value. Here again we emphasize the importance understanding and using the definition of absolute value. –(9 + x) = 3 – 2x 9 + x = –(3 – 2x) Same 9 + x = –(3 – 2x) 9 + x = 3 – 2x 10/2/2013 Absolute Value Equations Absolute Value Equations 10/2/2013

Absolute Value Equations Solving Absolute Value Equations Solve 9 + x= 3 – 2x 9 + x = –(3 – 2x) 9 + x = 3 – 2x 9 + x = 3 – 2x or 9 + x = –3 + 2x 2x + x = 3 – 9 or 9 + 3 = –x + 2x 3x = –6 or 12 = x Solving Absolute Value Equations: Example 4 (continued) Here we have eliminated duplicate equations and narrowed the search to the two equations 9 + x = 3 – 2x and 9 + x = –(3 – 2x) These two equations are equivalent to the original absolute value equation, so their solutions are the solutions of the original equation. Solving these equations yields two solutions, namely -2 and 12 and so the solution set for the original equation is {-2, 12}. In effect the equation with two absolute values is solved exactly like an equation with only one side in absolute value. Here again we emphasize the importance understanding and using the definition of absolute value. x = –2 or x = 12 Solution set is { –2, 12 } 10/2/2013 Absolute Value Equations Absolute Value Equations 10/2/2013

Solving Absolute Value Equations Think about it ! 10/2/2013 Absolute Value Equations Absolute Value Equations 10/2/2013