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.

 

 

 

 

010. `*   10

 

Question:  *   1.1.20 (was 1.1.12). Explain, step by step, how you solved the equation 5y + 6 = -18 - y

 

Your solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution:

* *  STUDENT SOLUTION WITH INSTRUCTOR COMMENT: 5y + 6 = 18 - y

 

Subtract 6 from both sides, giving us

 

5y = 12 - y

 

Add y to both sides,

 

5y + y = 12 or 6y = 12

 

divide both sides by 6

 

y = 2

 

INSTRUCTOR COMMENT: This is correct for equation 5y + 6 = 18 - y but the equation in the above note is 5y + 6 = -18 - y.

 

The solution to this equation is found by practically the same steps but you end up with y = -4.

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

 

Question:

1.1.38 \ 44 (was 1.1.30). Explain, step by step, how you solved the equation (2x+1) / 3 + 16 = 3x

 

Your solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution:

* *  STUDENT SOLUTION:

 

(2x + 1) / 3 + 16 = 3x

 

First, multiply both sides of the equation by 3

 

2x +1 + 48 =9x or

2x + 49 = 9x

 

subtract 2x from both sides to get

 

49 = 7x

 

Divide both sides by 7 to get

 

x = 7.

STUDENT QUESTION

 

I was wondering at the end since it ended up 49 = 7x and you divide by 7 and say x = 7…would you have to
make it a -7 if you move it to the opposite side of the equation?

INSTRUCTOR RESPONSE

 

It's not a matter of 'moving things around', but a matter of adding or subtracting the same quantity on both sides, or multiplying or dividing both sides by the same quantity.

In this case both sides are divided by 7, which doesn't involve any negative signs.

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

 

Question:

 

 

 

 

*   was 1.1.44 \ 36. Explain, step by step, how you solved the equation (x+2)(x-3) = (x+3)^2

 

Your solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution:

* *  STUDENT SOLUTION: (x+2)(x+3) = (x+3)^2

 

First, we use the distributive property to remove the parenthesis and get

 

x^2 - x - 6 = x^2 + 6x + 9

 

subtract x^2 from both sides,

 

-x - 6 = 6x + 9

 

Subtract 9 from both sides

 

- x - 6 - 9 = 6x or -x - 15 = 6x

 

add x to both sides

 

-15 = 7x

 

Divide both sides by 7

 

x = -15/7

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

 

Question:

 

*   1.1.52 (was 1.1.48). Explain, step by step, how you solved the equation x / (x^2-9) + 4 / (x+3) = 3 / (x^2-9).

 

Your solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution:

* *  Starting with

 

x / (x^2 -9) + 4 / (x+3) = 3 / (x^2 -9),

 

first factor x^2 - 9 to get

 

x / ( (x-3)(x+3) ) + 4 / (x+3) = 3 / ( (x-3)(x+3) ).

 

Multiply both sides by the common denominator ( (x-3)(x+3) ):

 

( (x-3)(x+3) ) * x / ( (x-3)(x+3) ) + ( (x-3)(x+3) ) * 4 / (x+3) = ( (x-3)(x+3) ) * 3 / ( (x-3)(x+3) ).

 

Simplify:

 

x + 4(x-3) = 3. 

 

Apply the Distributive Law, rearrange and solve:

 

x + 4x - 12 = 3

5x = 15

x = 3.

 

If there is a solution to the original equation it is x = 3.

 

STUDENT COMMENT

 

x / (x^2-9) + 4 / (x+3) = 3 / (x^2-9) Since you have like terms (x^2 – 9) on both sides, they cancel each other out

 

INSTRUCTOR RESPONSE

 

If something 'cancels' by multiplication or division, it has to 'cancel' from all terms. (x^2 - 9) is not a multiplicative or divisive factor of the term 4 / (x + 3) so that factor does not 'cancel'.

You can multiply or divide both sides by the same quantity, or add and subtract the same quantity from both sides.

Anything called 'cancellation' that doesn't result from these operations is invalid.

Because 'cancellation' errors are so common among students at this level, my solutions never mention anything called 'cancellation'.

 

If you multiply both sides of the equation

 

x / (x^2-9) + 4 / (x+3) = 3 / (x^2-9)

 

by (x^2 - 9), you get

 

( x / (x^2-9) + 4 / (x+3) ) * (x^2 - 9) = 3 / (x^2-9) * (x^2 - 9) so that

 

x / (x^2-9) * (x^2 - 9) + 4 / (x+3) * (x^2 - 9) = 3 / (x^2-9) * (x^2 - 9).  The (x^2 - 9) does then 'cancel' from two of the three terms, but not from the third.  You get

 

x + 4 / (x+3) * (x^2 - 9) = 3.

 

You're still stuck with an x^2 - 9 factor on one of the terms, and a denominator x - 3.

However this equation does represent progress. If you factor x^2 - 9 into (x-3)(x+3), things quickly simplify.

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

 

Question:

 

*   1.1.58 (was 1.1.54). Explain, step by step, how you solved the equation (8w + 5) / (10w - 7) = (4w - 3) / (5w + 7)

 

 

Your solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution:

* *  GOOD STUDENT SOLUTION:

 

1) clear the equation of fractions by multiplying both sides by the LCM (10w - 7)(5W + 7)

 

After cancellation the left side reads:

 

(5w+7)(8w + 5)

 

After cancellation the right side reads:

 

(10w - 7)(4w - 3)

 

multiply the factors on each side using the DISTRIBUTIVE LAW

 

Left side becomes: (40w^2) + 81w + 35

 

Right side becomes: (40w^2) - 58w + 21

 

3) subtract 40w^2 from both sides

add 58w to both sides

subtract 35 from both sides

Rewrite: 139w = - 14

 

Now divide both sides by 139 to get

 

w = - (14 / 139)

 

STUDENT QUESTION:

 

(5w+7)(8w+5) = (10w-7)(4w-3)
work what you can
40w^2 + 35 = 40w^2 +21
take away 40w^2 from both sides
didnt understand this one..;

INSTRUCTOR RESPONSE:

 

It doesn't look like you used the distributive law to multiply those binomials.

(5w+7)(8w+5) = 5w ( 8w + 5) + 7 ( 8w + 5)= 40 w^2 + 25 w + 56 w + 35 = 40 w^2 + 81 w + 35.
(10w-7)(4w-3) = 10 w ( 4 w - 3) - 7 ( 4 w - 3) = 40 w^2 - 30 w - 28 w + 21 = 40 w^2 - 58 w + 21.</h3>
 

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

 

Question: *   1.1.70 (was 1.1.78). Explain, step by step, how you solved the equation 1 - a x = b, a <> 0.

 

Your solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution:

* *  Start with

 

1 -ax = b, a <> 0.

 

Adding -1 to both sides we get

 

1 - ax - 1 = b - 1,

 

which we simplify to get

 

-ax = b - 1.

 

Divide both sides by -a, which gives you

 

x = (b - 1) / (-a). Multiply the right-hand side by -1 / -1 to get

x = (-b + 1) / a or

x = (1 - b) / a. **

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

 

Question:

*   extra problem (was 1.1.72). Explain, step by step, how you solved the equation x^3 + 6 x^2 - 7 x = 0 using factoring.

 

Your solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution:

* *  Starting with

 

x^3 + 6 x^2 - 7 x = 0 factor x out of the left-hand side:

x(x^2 + 6x - 7) = 0. Factor the trinomial:

x ( x+7) ( x - 1) = 0. Then

x = 0 or x + 7 = 0 or x - 1 = 0 so

x = 0 or x = -7 or x = 1. **

STUDENT QUESTION

 


I don’t understand this part of the equation x = 0 or x + 7 = 0 or x - 1 = 0 so
x = 0 or x = -7 or x = 1 ??? where do you get all of this from???

INSTRUCTOR RESPONSE


x ( x+7) ( x - 1) = 0

says that three different quantities, multiplied together, give you zero.

Now if three quantities multiplied together give you zero, what is the one thing you know for sure?

You know for sure that one of them is zero, because if you multiply three quantities that aren't zero you don't get zero (more specifically if you multiply three numbers, none of which are zero, you don't get zero).

The three quantities are x, x + 7 and x - 1. The only way you can get zero by multiplying these quantities is if one of them is zero.

So if

x ( x+7) ( x - 1) = 0

you know that x = 0, or x + 7 = 0, or x - 1 = 0.
 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

 

Question:

*   1.1.90 (was 1.2.18). The final exam counts as two tests.  You have scores of 86, 80, 84, 90.  What score do you need on the final in order to end up with a B average, which requires an average score of 80, and an A average, which requires a score of 90?

 

Your solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution:

* *  This can be solved by trial and error but the only acceptable method for this course, in which we are learning to solve problems by means of equations, is by an equation.

 

Let x be the score you make on the exam.

 

The average of the four tests is easy to find:

 

 

The final grade can be thought of as being made up of 3 parts, 1 part being the test average and 2 parts being the exam grade. We would therefore have

 

 

This gives us the equation

 

 

If the ave score is to be 80 then we solve

 

(85 + 2 * x) / 3 = 80.

 

Multiplying both sides by 3 we get

 

85 + 2x = 240.

Subtracting 85 from both sides we have

2 x = 240 - 85 = 155

so that

x = 155 / 2 = 77.5.

 

We can solve

 

(340 + x) / 5 = 90

 

in a similar manner. We obtain x = 92.5.

 

 

 

Alternative solution:

 

If we add 1/3 of the test average to 2/3 of the final exam grade we get the final average. So (using the fact that the test ave is 85%, as calculated above) our equation would be

 

1/3 * 85 + 2/3 * x = final ave.

 

For final ave = 80 we get

 

1/3 * 85 + 2/3 * x = 80.

 

Multiplying both sides by 3 we have

 

85 + 2 * x = 240.

 

The rest of the solution goes as before and we end up with

 

x = 77.5.

 

Solving 1/3 * 85 + 2/3 * x = 90 we get x = 92.5, as before. **

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

 

Question:

 

*   1.1.82 (was 1.1.90). Explain, step by step, how you solved the equation v = -g t + v0 for t.

 

 

 

Your solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Confidence Assessment:

Given Solution:

* *  NOTE: v0 stands for v with subscript 0; the whole expression v0 stands for the name of a variable. It doesn't mean v * 0.

 

Starting with v = -g t + v0, add -v0 to both sides to get

 

v - v0 = -gt.

 

Divide both sides by -g to get

 

(v - v0) / (-g) = t

 

so that

 

t = -(v - v0) / g = (-v + v0) / g.

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

Self-critique Rating:

 

*   Add comments on any surprises or insights you experienced as a result of this assignment.