Find x such that 3x²-7x+6<0

Question

This question belongs from rational inequalities.

Answer 1

Suppose we graphed the function 3x² - 7x + 6.  According to the inequality, we want to find x values in which the y value is below the x-axis or are negative.  If this was an equation, the values of x would have y values that are on the x-axis.

 

First, lets solve for the zeros by replacing the less than sign with an equal sign.  We will use these zeros later on.

 

3x² - 7x + 6 = 0

 

 

Use the quadratic formula to solve for the zeros.

 

x = (7 ± √(49 - 4(18))) / 6

x = (7 ± √(-23)) / 6

 

Note that we have a negative number under the square-root.  This means that we will have a complex number.  Complex numbers never touch the x-axis.  Also, since we have a positive parabola, this graph will never cross the x-axis, meaning that there are no values of x in which the y values are below the x-axis.

 

 

Another way to show that the graph is never below the x-axis we put this graph in vertex form.

 

y = a(x - h)² + k

 

where the vertex is (h, k).

 

y = 3(x² - (7/3)x + (49/36)) + (23/12)

 

y = 3(x - (7/6))² + (23/12)

 

h = 7/6

k = 23/12

 

The vertex is (7/6 , 23/12).  This is the minimum of the parabola.  This minimum is above the axis because the y value is positive.  Therefore, the graph never reaches below the x-axis.  Thus, there are no x values that make the inequality true.

Answer 2

3x²-7x+6<0

 

This is a quadratic equation in form ax²+ bx + c

This graphs to a parabola which opens upward since the squared term has a positive coefficient.

Let's find the vertex.  If the vertex is located above or on the x axis this will never be <0

If the vertex is below the x axis then the answer will be from the x coordinate of the vertex up to

the x axis

 

The x coordinate of the vertex is at x = -b/2a = -(-7)/[2(3)] = 7/6 

Find the value of y at x = 7/6

 

y = 3(7/6)²-7(7/6)+6 = 3(49/36)-49/6+ 6 = 49/12 - 98/12 + 72/12 = 23/12 or 1 11/12

 

Since the vertex is at (7/6 , 23/12) and opens upward this will never be <0 so the answer is the empty set