Solve the following inequality and write your answer using interval notation . Please show all your work

10^3 + 6x^2 - 90x - 54 < 0

I think you meant

10x^3 + 6x^2 - 90x - 54 < 0 , divide by 2
5x^3 + 3x^2 - 45x - 27 < 0
x^2(5x+3) - 9(5x+3) < 0
(5x+3)(x^2-9) < 0
(5x+3)(x+3)(x-3) < 0
critical values are x = -5/3, -3, and 3

Test the expression for different values, one in each part of the domain.
We don't have to find the actual value, just the sign of the answer
let x = -5 , for x < -5/3
- - - < 0 , that works
let x = -2 , for between -5/3 and -3
- + - > 0 , no good
let x = 0 , for between -3 and 3
+ + - < 0 , that works
let x = 10, for x > 3
+++ > 0 , no good

so x < -5/3 OR -3 < x < 3

I will let you change that to the interval notation that you were taught. Personally I prefer the above notation.

Me too but it gets counted wrong in this format.

To solve the inequality 10^3 + 6x^2 - 90x - 54 < 0, we need to find the values of x that make the inequality true.

Step 1: Simplify the inequality
10^3 can be simplified to 1000, so the inequality becomes:
1000 + 6x^2 - 90x - 54 < 0

Step 2: Combine like terms
Combine the terms -90x and -54 to get -144x:
1000 + 6x^2 - 144x < 0

Step 3: Set up a quadratic inequality
Rearrange the terms to form a quadratic equation in standard form and set it less than zero:
6x^2 - 144x + 1000 < 0

Step 4: Solve the quadratic equation
To find the values of x that satisfy the inequality, we can find the roots (or x-intercepts) of the quadratic equation. First, divide the entire equation by the common factor 2:
3x^2 - 72x + 500 < 0

Step 5: Factor or use the quadratic formula
Since the quadratic equation does not easily factor, we will use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation 3x^2 - 72x + 500 < 0, a = 3, b = -72, and c = 500.

Calculating the discriminant:
D = b^2 - 4ac
D = (-72)^2 - 4(3)(500)
D = 5184 - 6000
D = -816

Since the discriminant is negative, the quadratic equation has no real solutions. However, that does not mean the inequality has no solution.

Step 6: Determine the intervals
Since the quadratic equation has no real solutions, we can determine the intervals where the inequality is either greater than zero or less than zero by considering the leading coefficient, 3. Since 3 (a positive number) is positive, the parabola opens upward.

To find the intervals, we can either use a sign chart or graph the equation.

Using a sign chart:
Draw a number line and plot the critical points, which occur when the expression is equal to zero. The critical points are the x-values where the inequality might change.

In this case, the expression cannot be equal to zero since the quadratic equation has no real solutions. However, we can find the vertex of the parabola using the formula: x = -b / (2a)
x = -(-72) / (2 * 3)
x = 72 / 6
x = 12

So the critical point is x = 12.

Now, test the inequality on any value lower than 12, between 12 and any value higher than 12, and higher than 12, to determine the intervals where the inequality is either positive or negative.

Let's choose x = 0 (lower than 12):
3(0)^2 - 72(0) + 500 = 500 > 0

Let's choose x = 10 (between 12 and any value higher than 12):
3(10)^2 - 72(10) + 500 = 380 < 0

Let's choose x = 20 (higher than 12):
3(20)^2 - 72(20) + 500 = 500 > 0

From the sign chart or testing values, we have the following intervals:

Interval notation: (-∞, 12) U (12, ∞)
Explanation: The inequality 10^3 + 6x^2 - 90x - 54 < 0 is true for all x-values from negative infinity to 12 (excluding 12) and all x-values greater than 12.