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1. Solve the inequality.

6x^2 - x < 2

2.Graph the function. Show your work with the following: Fundamental Theorem of Algebra to know how many roots you’ll discover; Descartes    Rule of signs to determine how many will be positive, negative or imaginary; Rational Root Theorem to know the possible roots; and synthetic division with upper and lower bounds to find the roots.

f(x) = 3x^3 - 6x^2 + 3x + 6

Name any x-intercepts and y-intercepts in the graph of the function.

3.Graph the function.

f(x) = x^2 - 3x / 2x^2 - 3x - 9

Name any x-intercepts and y-intercepts in the graph of the function.

Name the coordinates of any holes in the graph of the function.

State the equations of any asymptotes in the graph of the function.

Sep 25, 2019

#1
+3

1.   6x^2 - x <  2        rearrange as

6x^2 - x - 2 <  0      factor the left side

(2x +1) (3x - 2) < 0

We have three possible  solution intervals   (-inf, -1/2)  , (-1/3, 2/3)  and ( 2/3, infinity)

Note that if   -1/2 < x < 2/3...  this inequality will be true

So......the solution interval  is   ( -1/2, 2/3)

On a number line, draw open circles at  -1/2  and 2/3    and connect these   Sep 25, 2019
#2
+2

2.    f(x)  =  3x^3 - 6x^2 + 3x + 6

Here's the graph :  https://www.desmos.com/calculator/68wqsc5sft

Possible positive roots  ..  f(x) has  two sign changes so   two  or zero  possible positive roots

f(-x)  = -3x^3 - 6x^2 - 3x + 6  =  1 sign change  so  one or zero posiible negative roots

Rational Roots Theorem  shows that the possible rational  roots are   ± [ 1/3, 2/3, 1, 2, 3, 6 ]

Upper Bound

2 [ 3    -6    3     6  ]              3  [  3     -6    3      6  ]

6     0    6                                9    9    36

______________                ________________

3     0     3    12                      3     3    12    42

All the signs on the bottom row  are positive whne x  = 3.....so this is the upper bound on zeroes

Lower Bound

-1  [ 3   -6    3    6  ]

-3    9   -12

______________

3    -9   12   -6

We have alternating signs on the bottom row.....therefore....x =-1  is the lower bound on zeroes

The y intercept  is  (0,6)

The x intercept  (and the only real root)  is  x ≈ -.696   Sep 25, 2019
edited by CPhill  Sep 25, 2019
#3
+2

3.

Here's the graph :  https://www.desmos.com/calculator/zgv7tpvl0y

x^2 - 3x                     x ( x - 3)                   x

___________   =     _____________  =   ______

2x^2 - 3x - 9              (2x + 3) (x - 3)           2x  + 3

The vertical asynptote  is the x values that make the  denominator = 0

These are   x  = -3/2

The  horizontal asymptote  is y  = 1/2   [the ratio of the coefficients on the variable x in the numerator/denominator ]

The "hole"  occurs at    x - 3  =  0     ⇒   x  = 3

And the associated y value is   3 / {2(3) + 3]  =  3 / 9  = 1/3

So...the hole is at (3, 1/3)

The graph shows that the x and y intercepts occur at  (0,0)   Sep 25, 2019