If your solution to stated problem does not match the given solution, you should self-critique per instructions at http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm.

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.

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Question: *   1.6.12 (was 1.6.6). Explain how you found the real solutions of the equation | 1 - 2 z | + 6 = 9.

Your solution:

Confidence Assessment:

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):

Self-critique Rating:

Question: *   1.6.30 (was 1.6.24). Explain how you found the real solutions of the equation | x^2 +3x - 2 | = 2

Your solution:

Confidence Assessment:

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):

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.

Your solution:

Confidence Assessment:

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):

Self-critique Rating:

Question: *   1.6.52 \ 48 (was 1.6.42). Explain how you found the real solutions of the inequality | -x - 2 | >= 1.

Your solution:

Confidence Assessment:

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):

Self-critique Rating: