Step 4: Solve both equations. b + 2 = +(b 3) b + 2 = 3) – –(b – b + 2 = b - 3 b + 2 = + 3 –b 2 = 2b + 2 = 3 –3 This is false. 2b = 1 There is no b = solution from this equation. Since the original equation is satisfied even if only of the two equations in Step 3 were satisfied, then this problem has a solution: b = . This value of b will make the original equation true. Example 5: Solve |x 4| = |4 x|. – – Step 1: Let the expression on one Done, because the expression on the side of the equation consist only of a left already consists only of a single single absolute value expression. absolute value expression. Step 2: Is the number on the other No, because |4 x| is surely not – side of the equation negative? negative (the absolute value of a number can never be negative). Proceed to Step 3. Step 3: To satisfy the equation, the x 4 = +(4 x) x 4 = x) – – – –(4 – expression inside the first absolute value, x 4, can either be equal to – +(4 x) or x). These correspond – –(4 – to two equations. [Notice the similarity to Step 3 of Example 1.] Step 4: Solve both equations. x 4 = +4 x x 4 = x) – – – –(4 – 2x 4 = 4 x 4 = + x – – –4 2x = 8 = –3 –3 x = 4 This is true no matter what value x is. All real numbers are solutions to this equation. Since the original equation is satisfied even if only of the two equations in Step 3 were satisfied, then the solution to the absolute value equation is the set of all real numbers.
Step 4: Solve both equations. b + 2 = +(b – 3) b + 2 = b - 3 2 = –3 This is false. There is no solution from this equation. b + 2 = –(b – 3) b + 2 = –b + 3 2b + 2 = 3 2b = 1 b = Since the original equation is satisfied even if only of the two equations in Step 3 were satisfied, then this problem has a solution: b = . This value of b will make the original equation true. Example 5: Solve |x – 4| = |4 – x|. Step 1: Let the expression on one side of the equation consist only of a single absolute value expression. Done, because the expression on the left already consists only of a single absolute value expression. Step 2: Is the number on the other side of the equation negative? No, because |4 – x| is surely not negative (the absolute value of a number can never be negative). Proceed to Step 3. Step 3: To satisfy the equation, the expression inside the first absolute value, x – 4, can either be equal to +(4 – x) or –(4 – x). These correspond to two equations. [Notice the similarity to Step 3 of Example 1.] x – 4 = +(4 – x) x – 4 = –(4 – x) Step 4: Solve both equations. x – 4 = +4 – x 2x – 4 = 4 2x = 8 x = 4 x – 4 = –(4 – x) x – 4 = –4 + x –3 = –3 This is true no matter what value x is. All real numbers are solutions to this equation. Since the original equation is satisfied even if only of the two equations in Step 3 were satisfied, then the solution to the absolute value equation is the set of all real numbers.
7-math-lm_mod3.docx