Solve each inequality by using algebra.

8. x 2 + 10x + 1 ≥ 12
9. x 2 + 13x + 45 < 5
10. -2x 2 + 3x + 12 > 10
11. Business A consultant advises the owners of a beauty salon that their profit p each month can be modeled by p(x) = -50x 2 + 3500x - 2500, where x is the average cost that a customer is charged. What range of costs will bring in a profit of at least $50,000?

one-ninth of polymer plastics' sales made in New England. New England sales amount to 6000,000, what are the total sales of the company

To solve each inequality, we will follow a general process using algebraic techniques. Here's how you can solve each inequality step by step:

8. x^2 + 10x + 1 ≥ 12:
First, subtract 12 from both sides to isolate the quadratic term:
x^2 + 10x - 11 ≥ 0
Next, factor the quadratic equation. Since this quadratic cannot be factored easily, you can use the quadratic formula or complete the square if necessary.
After finding the critical points (where the quadratic is equal to zero), you can determine the sign of the quadratic expression in different intervals and accordingly find the solution to the inequality.

9. x^2 + 13x + 45 < 5:
Similarly, subtract 5 from both sides:
x^2 + 13x + 40 < 0
Next, factor the quadratic equation:
(x + 8)(x + 5) < 0
Set each factor less than zero and solve for x:
x + 8 < 0 and x + 5 < 0
Solving these inequalities, you will find the range of x values where the quadratic is less than zero.

10. -2x^2 + 3x + 12 > 10:
Subtract 10 from both sides:
-2x^2 + 3x + 2 > 0
Factor the quadratic equation:
(-2x - 1)(x - 2) > 0
Set each factor greater than zero and solve for x:
-2x - 1 > 0 and x - 2 > 0
Solving these inequalities, you will find the range of x values where the quadratic is greater than zero.

11. To find the range of costs (x) that will bring in a profit of at least $50,000 (p ≥ 50000), we need to solve the inequality:

-50x^2 + 3500x - 2500 ≥ 50000

First, subtract 50000 from both sides:
-50x^2 + 3500x - 52500 ≥ 0

Next, rearrange the equation:
-50x^2 + 3500x - 52500 + 52500 ≥ 0 + 52500

Simplifying:
-50x^2 + 3500x ≥ 52500

Divide the entire inequality by -50 (Note: when dividing by a negative number, the inequality sign flips):
x^2 - 70x ≤ -1050

To solve this quadratic inequality, set it equal to zero:
x^2 - 70x + 1050 ≤ 0

Next, factor the quadratic equation:
(x - 30)(x - 35) ≤ 0

Set each factor less than or equal to zero:
x - 30 ≤ 0 and x - 35 ≤ 0

Solve these inequalities to find the range of costs that will bring in a profit of at least $50,000.