How do you solve the inequality #6x^2-5x6#?

Question

How do you solve the inequality #6x^2-5x>6#?

Lily Anderson · Accepted Answer

I would start by solving it as a normal equation (2nd degree). So you write:
#6x^2-5x-6>0# (I took the #6# to the left);
Solving your equation you should get two values #x_1=3/2# and #x_2=-2/3#.
Now the tricky bit:
your inequality asks you for values of #x# that make your equation have a value bigger than zero.
They cannot be #x_1 and x_2# because at these point your equation IS equal to zero.
Graphically your function gives the following parabola:

graph{6x^2-5x-6 [-11.96, 13.02, -7.14, 5.34]}

So, basically I have to choose values that are outside the boundaries formed by the two values #x_1 and x_2# to get a value bigger than zero (in the graph the two bits that are ABOVE the #x# axis).!!!
Consider #x_1=3/2=1.5# ok I cannot choose it but what about #2# (which is bigger)?
If I put #x=2# in the equation I get: #6*4-5*2-6=8>0# YES!
Consider now #x_2=-2/3=-0.67# again I cannot choose it but what about #-1#?
If I put #x=-1# in the equation I get: #6*(-1)^2-5*(-1)-6=5>0# YES!
So, OUTSIDE the interval bound by #x_1 and x_2# you can choose #x#.
You express this by writing your solution as:
#-2/3>x>3/2#
Or graphically

Hope it helps

Serenity Jones · Answer

undefined To solve the inequality $6x^2 - 5x > 6$, you first need to rearrange it to form a quadratic inequality in standard form. This means bringing all terms to one side of the inequality sign and setting the expression equal to zero.

$6x^2 - 5x - 6 > 0$

Now, you have a quadratic inequality in the form $ax^2 + bx + c > 0$.

To solve this type of quadratic inequality, you can use various methods, such as factoring, graphing, or the quadratic formula. One common approach is to find the critical points by solving the related equation $6x^2 - 5x - 6 = 0$ and then test intervals to determine where the inequality holds true.

To find the critical points, you can use the quadratic formula:

$x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}$

Substitute $a = 6$, $b = -5$, and $c = -6$ into the quadratic formula and solve for $x$:

$x = \frac{{-(-5) \pm \sqrt{{(-5)^2 - 4 \cdot 6 \cdot (-6)}}}}{{2 \cdot 6}}$

After calculating, you'll find two solutions for $x$.

Once you have the critical points, you can use them to create intervals on the number line and test each interval to see where the inequality holds true. You can use test points within each interval to determine whether the expression is greater than zero or not. The intervals where the expression is greater than zero will be the solution to the inequality.

How do you solve the inequality #6x^2-5x&gt;6#?

How do you solve the inequality #6x^2-5x>6#?