x^2-6>-5x

6x^2-5x+1<0
Solve (using any method), write your answer in interval notation and graph the solution set.

1. x^2 + 5x - 6 > 0

(x+6)(x-1) > 0
critical values are x = -6 and x = 1

so the parabola y = x^2 + 5x - 6 crosses the x-axis at -6 and 1, and knowing about the basic shape, (it opens upwards), we can state that it will be above for
x < -6 OR x > 1

2. 6x^2 - 5x + 1 < 0
(6x - 1)(x + 1) < 0
critical values are .....
since we want to be below the x-axis ( < 0)
.......

take it from here, and use the notation that was taught to you,

[1/3, 1/2]

To solve these inequalities and represent the solution set in interval notation, we will follow these steps:

1. Solve the quadratic inequalities.
2. Write the solution sets in interval notation.
3. Graph the solution sets on a number line.

Let's solve the first quadratic inequality: x^2 - 6 > -5x.

Step 1: Solve the quadratic inequality:
Rearrange the inequality to the standard quadratic form: x^2 + 5x - 6 > 0.
Factorize the quadratic equation: (x + 6)(x - 1) > 0.

Next, determine the sign of the quadratic expression by considering the sign of each factor or by using a sign chart:

Case 1: (x + 6) > 0 and (x - 1) > 0
For (x + 6) > 0, x > -6.
For (x - 1) > 0, x > 1.
Thus, the solution set for this case is x > 1.

Case 2: (x + 6) < 0 and (x - 1) < 0
For (x + 6) < 0, x < -6.
For (x - 1) < 0, x < 1.
Thus, there is no solution in this case.

Step 2: Write the solution set in interval notation:
Based on the two cases above, the solution set is: (-∞, -6) U (1, ∞).

Now, let's solve the second quadratic inequality: 6x^2 - 5x + 1 < 0.

Step 1: Solve the quadratic inequality:
Rearrange the inequality to the standard quadratic form: 6x^2 - 5x + 1 < 0.

Next, determine the sign of the quadratic expression by considering the sign of each factor or by using a sign chart. However, this quadratic expression does not factorize easily since the discriminant is negative. Therefore, we will use an alternate method such as completing the square or the quadratic formula.

Using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a)
For the quadratic expression 6x^2 - 5x + 1, the coefficients are: a = 6, b = -5, and c = 1.

By substituting these values into the formula, we get:
x = (-(-5) ± √((-5)^2 - 4 * 6 * 1)) / (2 * 6)
x = (5 ± √(25 - 24)) / 12
x = (5 ± 1) / 12

Therefore, x can be either (5 + 1) / 12 = 6 / 12 = 1/2 or (5 - 1) / 12 = 4 / 12 = 1/3.

Step 2: Write the solution set in interval notation:
Based on the solutions from the quadratic formula, the solution set is: (1/2, 1/3).

Now, let's graph the solution sets on a number line:

On a number line, mark the points: -6, 1/2, and 1/3.
The solution sets can be represented as shaded intervals as follows:
(-∞, -6) U (1, ∞) for the first quadratic inequality, and
(1/2, 1/3) for the second quadratic inequality.

Graph the shaded intervals on the number line accordingly:
-----------------------------------------------------
-∞ ----|----|---(-6)---|-(-1/3)---|(1/2)---|(1)---|---- ∞
(-∞, -6) (1/2, 1/3)

The solution sets are represented on the number line as the shaded intervals.