(64)A common mistake when solving equations is the following:
The equation: 2(x – 2) = x + 3
First step in solving: 2x – 2 = x + 3
Write a clear explanation of what error has been made. 
What could be done to avoid this error?

If we have this equation, we must remove the parentheses
by using the distributive principle

2(x - 2) = x + 3

The distributive principle says you must multiply the outer 
red 2 by both the blue x and the green -2, so that the first 
step in solving is

2x - 4 = x + 3

The mistake was forgetting to multiply the red 2 by the green -2.
Nobody forgets to multiply the red 2 by the blue x, but then people
get in a rush and forget to multiply the red 2 by the green -2, and
they get

2x - 2 = x + 3

instead, which is wrong.

One way to avoid making this error is to get in the habit of drawing
lines something like this to remind you that you must make BOTH
multiplications, not just the first one only:
 _
| |
2(x – 2) = x + 3
|____|

 
(92) Statistics. Sam must have an average of 70 or more in his summer
course to obtain a grade of C. His first three test grades were 
75, 63, and 68. Write an inequality representing the score that Sam 
must get on the last test to get a C grade. 

Let the fourth grade be x, then the average is found by adding up
the four grades and dividing by 4, like this

 75 + 63 + 68 + x
------------------  
         4     

This must be greater than or equal to a 70 in order for Sam to make
a C.  So we form the inequality

 75 + 63 + 68 + x
------------------ > 70  
         4

or

 206 + x
--------- > 70  
    4 
         
Then we multiply both sides of the inequality by 4 to clear of
fractions.  The inequality symbol will not reverse because we
are multiplying by a positive number 4.

206 + x > 280

Subtract 206 from both sides:

      x > 74

(94) Business and finance. The cost for a long-distance telephone call 
is $0.36 for the first minute and $0.21 for each additional minute or 
portion thereof. Write an inequality representing the number of 
minutes a person could talk without exceeding $3.

Let x = the number of minutes:

The first minute costs $0.36
The remaining x-1 minutes cost $0.21 each
So the remaining x-1 minutes altogether costs $0.21(x-1)

Total cost = $0.36 + $0.21(x-1) 

This must be less than or equal $3.00. So we have:

$0.36 + $0.21(x-1) < $3.00

Multiply through by 100 and drop the $'s

36 + 21(x - 1) < 300

 12 + 7(x - 1) < 100 

   12 + 7x - 7 < 100 

        7x + 5 < 100

Subtract 5 from both sides

            7x < 95

Divide both sides by 7, which will not
reverse the inequality symbol:

             x < 13.57142857

So the answer is really

             x < 13 minutes

which will be 36 cents for the first minute
and 21 cents for the other 12 minutes, and
will cost Sam $2.88.  However if Sam talks even
one second more than 13 minutes, it will cost
him another 21 cents. making the bill be
$3.09 which exceeds $3.00

Edwin