Your solution, attempt at solution: If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it.This response should be given, based on the work you did in completing the assignment, before you look at the given solution. 014. `* 14 ********************************************* Question: * 1.6.12 (was 1.6.6). Explain how you found the real solutions of the equation | 1 - 2 z | + 6 = 9. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: To solve this we first need to subtract 6 from both sides giving the absolute value of 1-2x=3 but we understand that the absolute value sign means that it’s really a measure of distance from zero meaning that the problem becomes 1-2x=3 and 1-2x=-3. Now we subtract a one form both sides giving -2x=2 and -2x=4 and then we divide by a negative 2 to isolate the x variable giving x=-1 and x=2 so our solution is (-1,2). confidence rating #$&*:ok ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: * * Starting with | 1-2z| +6 = 9 we add -6 to both sides to get | 1 - 2z| = 3. We then use the fact that | a | = b means that a = b or a = -b: 1-2z=3 or 1-2z= -3 Solving both of these equations: -2z = 2 or -2z = -4 we get z= -1 or z = 2 We express our solution set as {-1, 2} ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok ------------------------------------------------ Self-critique Rating:ok ********************************************* Question: * 1.6.30 (was 1.6.24). Explain how you found the real solutions of the equation | x^2 +3x - 2 | = 2 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: To solve this problem we need to understand that the absolute value really means the distance from zero the number value is located, meaning that this can be x^2+3x-2=2 or x^2+3x-2=-2. Now we solve these by setting them equal to zero and then factoring. In the first we subtract a 2 from both sides and then factor the trinomial. Giving (1,-4) for the first, and (0,-3) for the second. confidence rating #$&*:ok ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: * * My note here might be incorrect. If the equation is | x^2 +3x -2 | = 2 then we have x^2 + 3x - 2 = 2 or x^2 + 3x - 2 = -2. In the first case we get x^2 + 3x - 4 = 0, which factors into (x-1)(x+4) = 0 with solutions x = 1 and x = -4. In the second case we have x^2 + 3x = 0, which factors into x(x+3) = 0, with solutions x = 0 and x = -3. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok ------------------------------------------------ Self-critique Rating: ********************************************* Question: * 1.6.40 \ 36 (was 1.6.30). Explain how you found the real solutions of the inequality | x + 4 | + 3 < 5. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: To solve this we first subtract 3 from both sides giving x+4<2 next we add the opposite to the inequality so -2
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Given Solution: * * STUDENT SOLUTION: | x+4| +3 < 5 | x+4 | < 2 -2 < x+4 < 2 -6 < x < -2 STUDENT QUESTION I was hoping to see more in the given solution as to why we move 2 to the left of the inequality. I think there is a formula for that, but I don’t remember what it is. Could you explain why we move the 2? INSTRUCTOR RESPONSE The 2 doesn't get moved. To understand what's going on: Think about the inequality | A | < = 4. This is clearly true if A = 4, 3, 2, 1 or 0. It's also clearly true if A = -1, -2, -3 or -4. It's not true if A = -5 or -6 or -7, etc.. So | A | < = 4 means the same thing as -4 <= A <= 4. More generally | A | < B says the same thing as - B < A < B. In your solution you said that | x + 4 | + 3 < 5 add -3 to both sides give us x + 4 < 2 This isn't so. The | | signs don't go away when you add -3 to both sides. You get | x + 4 | < 2, which means the same thing as -2 < x + 4 < 2 because of the rule we just say, that | A | < B means -B < A < B. Correcting your solution: | x + 4 | + 3 < 5 add -3 to both sides | x + 4 | < 2 add -2 to the left of the inequality -2 < x + 4 < 2 apply the rule for | A | < B with A = x + 4 and B = 2 -2-4 < x+4-4 < 2-4 simplify to get -6 < x < -2 &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):ok ------------------------------------------------ Self-critique Rating:ok ********************************************* Question: * 1.6.52 \ 48 (was 1.6.42). Explain how you found the real solutions of the inequality | -x - 2 | >= 1. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: To solve this we need to understand that the absolute value will present us with two forms, these forms are -x-2>=1 and -x-2<=-1. Once we are to this form we can solve them like a linear equation. Now we add a negative 2 to both sides giving us -x>=3 and -x<=1. Now we can divide by a -1 giving x<=-3 and x>=-1. confidence rating #$&*:ok ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: * * Correct solution: | -x -2 | >= 1 Since | a | > b means a > b or a < -b (note the word 'or') we have -x-2 >= 1 or -x -2 <= -1. These inequalities are easily solved to get -x >= 3 or -x <= 1 or x <= -3 or x >= -1. So our solution is {-infinity, -3} U {-1, infinity}. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):I can see that I left out a step of the problem because I did not put it in interval notation. ------------------------------------------------ Self-critique Rating:2