These questions are pertaining to College Algebra

Question

Write a statement using an absolute value equation or inequality

 

a) q is no more than 8 units from 22

 

b) When a model kite was flown in crosswinds in tests to determine its limits of power extraction, it attained speeds of 98 to 148 ft per sec in winds of 16 to 26 ft per sec. Using x as the variable in each case, write absolute value inequalities that corresponds to these ranges.

 

c) | x^2 + 2  | -      11     = 0

         x                     3

Answer 1

(a). The statement "q is no more than 8 units from 22" can be expressed       as the absolute value inequality lq-22l≤8.

 

      If lq-22l≤8, then -8≤q-22≤8

 

                   So, 14≤q≤30

 

(b). Let x = kite speed

 

      The midpoint of the interval [98, 148] is 123 and half the length of       the interval is 25.

 

      So, in terms of an absolute value inequality, we have lx-123l≤25.

 

         Check:  if lx-123l≤25, then -25≤x-123≤25

 

                                          So, 98≤x≤148

 

      Let x = wind speed

 

      The midpoint of the interval [16,26] is 21 and half the length of           the interval is 5.

 

      So, the absolute value inequality that models this situation is 

      lx-21l≤5.

Answer 2

a)  q is no more than 8 units from 22

     22 + 8 = 30

     22 - 8 = 14

     q has to be on the interval [14, 30]

      14 ≤ q ≤ 30

 

b)  There are two variables involved here...

     Velocity and windspeed

 

Since our velocity v is a function of windspeed w

Domain of w is [16 ft/sec, 26 ft/sec]

with a range v of [98 ft/sec, 148 ft/sec]

 

16 ≤ w ≤ 26

98 ≤ v ≤ 148

 

You can use "x" instead of v to indicate velocity if you desire

 

c)  | x² + 2 | -  11 = 0

 

|x² + 2| = 11

 

x² + 2 = 11     or     x²+2 = -11

x² = 9                     x² = -13

x = ±3                   Does not exist

 

x = {-3, 3}

    

Answer 3

Hi Keissa

 

(a)  The symbol for "no more than" is "<" or "less than". Then the statement: q is no more than 8 units from 22 is indicating that the variable "q" is placed between 22 and 22+8. In other words, we know that q is greater than 22, but lower than 22 plus 8 units above (= 30 units).

 

It can be expressed by the following inequality:

 

22 < q < 30 

 

(b) If x represents the velocity, we have:  98 ft/s < x < 148 ft/s

 

(c) For resolving X from this absolute equation you have to decompose the absolute value in its positive and negative parts:

 

Positive part:   | x² + 2| = x² + 2

 

x² + 2 -11 = 0

 

x² = 9        x = -9   and x = 9

 

Negative part:  | x² + 2 | = -( x² + 2)  = - x² - 2

 

- x² -2 - 11 = 0

 

- x² = 13     or      x² = - 13    No solution because never a squared real number is negative

 

Then, the solution of (c)  is  x = 3  and x = -3

 

I hope it helped you. Good luck with your scores.