How do you solve this inequality?

Question

How do you solve this inequality?

 

2x³-9x²+8x+9<x+3

Answer 1

Combine all the terms on the left hand side of the inequality:

 

• 2x³-9x²+8x+9<x+3            the original inequality becomes
• 2x³-9x²+7x+6<0                "converted polynomial"

 

You must determine a root of this polynomial. To do so use the Rational Root Test to determine all the possible roots. Possible roots are determined by dividing all the factors of the constant (6) by all the factors of the leading coefficient (2) and assigning a plus or minus value to each.

 

• All the factors of the constant are: 6, 3, 2, 1
• All the factors of the leading coefficient are: 2,1

Possible Roots are: 6/2, 6/1, 3/2, 3/1, 2/2, 2/1/, 1/1 each plus or minus.

From this list try 6/2 = 3/1 = 3. Use Synthetic Division to confirm that it is a root of the "converted polynomial".

 

                             3l  2   -9   7   6

 

                                  

                                        6 -9  -6

                              ____________

                                 2    -3 -2    0

 

Because the remainder is zero, 3 is a root of the converted polynomial which gives a binomial of (x-3). Thus:

 

                    (2x²-3x-2)(x-3) = 2x³-9x²+7x+6

 

Now factor this quadratic (2x²-3x-2). When you do you will find 2 binomials (2x+1)(x-2). Setting all binomial equal to zero you will find that the solution is:

 

 x=3 or x=-1/2 or x=2.

 

 

 

Answer 2

2x³ - 9x² + 8x + 9 < x + 3

2x³ - 9x² + 7x + 6 < 0

Using Rational Root Theorem and Synthetic Substitution

 

3 | 2  -9  7  6

          6 -9 -6

2 | 2  -3 -2

          4  2 

     2   1  0

 

Factors are (x - 3)(x - 2)(x + 0.5)

 

From here determine for which values of x the product is < 0.

Answer 3

Bring all terms to the left side to make the right side equal to zero.

 

2x³ - 9x² + 7x + 6 < 0

 

 

Then, we can factor the left side using synthetic division.

 

3 | 2     -9      7       6

 

             6     -9      -6

___________________

     2     -3     -2       0

 

 

(x - 3)(2x² - 3x - 2) < 0

 

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

 

 

Now here is the trick.  I will turn this "less than" sign into an equal sign.  This allow us to find the zeros.  These are the values of x where the graph is on the x-axis.  The zeros are then

 

x = -1/2             x = 2            x = 3

 

 

Since this is a positive cubic function, imagine how the graph will look like.  The intervals of x where the graph is below the x-axis is then

 

(-∞, -1/2)∪(2, 3)