Step 3: If the absolute value of an expression is equal to a positive number, say a, then the expression inside the absolute value can either be a or –a (Again, think, why?). Equate the expression inside the absolute value sign to a and to and solve both equations. –a, Example 1: Solve |3a 4| 9 = 15. – – Step 1: Let the expression on one side |3a 4| 11 = 15 – – of the equation consist only of a single |3a 4| = 26 – absolute value expression. Step 2: Is the number on the other side a positive number, 26, No, it’s of the equation negative? so proceed to step 3 Step 3: To satisfy the equation, the 3a 4 = 26 3a 4 = – – –26 expression inside the absolute value can either be +26 or These correspond –26. to two equations. Step 4: Solve both equations. 3a 4 = 26 3a 4 = – – –26 3a = 30 3a = –22 a = 10 a = We can check that these two solutions make the original equation true. If a = 10, then |3a 4| 9 = |3(10) 4| 9 = 26 9 = 15. Also, if a = then |3a 4| – – – – – –22/3, – – 9 = |3(–22/3) 4| 9 = |–26| 9 = 15. – – – Example 2: Solve |5x + 4| + 12 = 4. Step 1: Let the expression on one |5x + 4| + 12 = 4 side of the equation consist only of a |5x + 4| = single absolute value expression. –8 Step 2: Is the number on the other Yes, a negative number, There it’s –8. side of the equation negative? is no solution because |5x + 4| can never be negative, no matter what we substitute for x.
Step 3: If the absolute value of an expression is equal to a positive number, say a, then the expression inside the absolute value can either be a or –a (Again, think, why?). Equate the expression inside the absolute value sign to a and to –a, and solve both equations. Example 1: Solve |3a – 4| – 9 = 15. Step 1: Let the expression on one side of the equation consist only of a single absolute value expression. |3a – 4| – 11 = 15 |3a – 4| = 26 Step 2: Is the number on the other side of the equation negative? No, it’s a positive number, 26, so proceed to step 3 Step 3: To satisfy the equation, the expression inside the absolute value can either be +26 or –26. These correspond to two equations. 3a – 4 = 26 3a – 4 = –26 Step 4: Solve both equations. 3a – 4 = 26 3a = 30 a = 10 3a – 4 = –26 3a = –22 a = We can check that these two solutions make the original equation true. If a = 10, then |3a – 4| – 9 = |3(10) – 4| – 9 = 26 – 9 = 15. Also, if a = –22/3, then |3a – 4| – 9 = |3(–22/3) – 4| – 9 = |–26| – 9 = 15. Example 2: Solve |5x + 4| + 12 = 4. Step 1: Let the expression on one side of the equation consist only of a single absolute value expression. |5x + 4| + 12 = 4 |5x + 4| = –8 Step 2: Is the number on the other side of the equation negative? Yes, it’s a negative number, –8. There is no solution because |5x + 4| can never be negative, no matter what we substitute for x.
7-math-lm_mod3.docx