Interval Notation

1.  Solve and express your solution set in interval notation.

 

|(2x-1)/(3)|>=(5)/(6)

 

Remove the absolute value term.  This creates a \ on the right-hand side of the equation because |x|=\x.

(2x-1)/(3)>=\((5)/(6))

 

Set up the + portion of the \ solution.

(2x-1)/(3)>=(5)/(6)

 

Solve the first equation for x.

x>=(7)/(4)

 

Set up the – portion of the \ solution.  When solving the – portion of an inequality, flip the direction of the inequality sign.

(2x-1)/(3)<=-((5)/(6))

 

Multiply each term in the inequality by 3.

2x-1<=-3((5)/(6))

 

Multiply -3 by each term inside the parentheses.

2x-1<=-(5)/(2)

 

Move all terms not containing x to the right-hand side of the inequality.

2x<=-(3)/(2)

 

Divide each term in the inequality by 2.

x<=-(3)/(4)

 

The solution to the inequality includes both the positive and negative versions of the absolute value.

x>=(7)/(4) or x<=-(3)/(4)

 

3.   Under certain conditions, the power P, in watts per hour generated is by a

windmill with winds blowing v miles per hour is given by

=  .

How fast must the wind blow in order to generate 120 watts of power in 1 hr?

 

the power P, in watts per hour, generated by a windmill with winds blowing v miles per hour is given by
P(v)= 0.015v^3
 

 

 

120 = 0.015v^3
120/0.015 = v^3
v = 20 mph

 

6.    Sketch the graph of the function,

 

 

7.   Find a rational function that satisfies the given conditions.

Vertical asymptotes x  =  -3  and x  =  5;      x-intercept  (-1, 0)

 

Vertical asymptotes x  =  -3  and x  =  5

Means 

 

The   denominator will be = (x+3)(x+5)

 

X intercept = (1,0)  so (x-1)

 

Equation = a*(x-1)/( x+3)(x+5)

 

 

 

8.  The frame of a picture is 28 cm by 32 cm outside and is of uniform width. What

is the width of the frame if 192  of the picture shows?

 

 

 

The shape of the picture is a rectangle formed by removing a constant from both

 

Dimensions of the outer frame is  28 x 32. 


(28 – c)(32 – c) = 192
896 – 60c + c^2 = 192
c^2 – 60c + 704 = 0

 

C= 16 , 44

 

So  the dimensions are 12,16.

 

C is twice the frame so width of frame= 16/2 = 8 cm

 

 

9.    Solve the inequality and write the solution set in interval notation.

 

|(x+3)/(x-4)|<2

 

Remove the absolute value term.  This creates a \ on the right-hand side of the equation because |x|=\x.

(x+3)/(x-4)<\(2)

 

Set up the + portion of the \ solution.

(x+3)/(x-4)<2

 

Solve the first equation for x.

x<4 or x>11

 

Set up the – portion of the \ solution.  When solving the – portion of an inequality, flip the direction of the inequality sign.

(x+3)/(x-4)>-(2)

 

To set the left-hand side of the inequality equal to 0, move all the expressions to the left-hand side.

(x+3)/(x-4)+(2)>0

 

Remove the parentheses around the expression 2.

(x+3)/(x-4)+2>0

 

Multiply each term by a factor of 1 that will equate all the denominators.  In this case, all terms need a denominator of (x-4).

(x+3)/(x-4)+2*(x-4)/(x-4)>0

 

Multiply the expression by a factor of 1 to create the least common denominator (LCD) of (x-4).

(x+3)/(x-4)+(2(x-4))/(x-4)>0

 

The numerators of expressions that have equal denominators can be combined.  In this case, ((x+3))/((x-4)) and ((2(x-4)))/((x-4)) have the same denominator of (x-4), so the numerators can be combined.

((x+3)+(2(x-4)))/(x-4)>0

 

Combine all similar expressions in the polynomial.

(3x-5)/(x-4)>0

 

Since this is a ‘greater than 0’ inequality, all intervals that make the expression positive are part of the solution.

x<(5)/(3) or x>4

 

The solution to the inequality includes both the positive and negative versions of the absolute value.

x<4 or x<(5)/(3)

 

 

10.

 

The function,   , gives the height s, in feet, of an

object thrown from a cliff that is 1920 ft high. Here t is the time, in seconds,

that the object is in the air. For what times is the height less than 640ft?

 

 

for time  is the height less than 640 we have to equate

-16*t^2+32*t+1920 < 640

-16*t^2+32*t+1280 < 0

 

-t^2-2*t+80 < 0

 

(t-1)^2 > 81

 

t-1 > 9

 

t > 10

 

For times more than 10 sec.