Math Is Fun Forum

jadewest

Member

Registered: 2021-02-20

Posts: 44

Solving Quadratic Inequalities

Hello,

I can't solve this problem:

Nathan is a product manager for a home security company.  Through testing and focus groups, he has modeled the equation P = -(x - 32)2 + 1000 to represent the profit (in thousands) of the price x for a new home security camera.  Find the prices in dollars at which Nathan can make a profit of at least $471, 000.  Show your work as it is done in the lesson content and include units of measurement..  HINT:  Use 471 as part of the inequality you set up, not 471,000.

Thank you so much,
Jade

Bob

Administrator

Registered: 2010-06-20

Posts: 10,140

Re: Solving Quadratic Inequalities

hi Jade,

Let's ignore the inequality for the moment and find instead when -(x - 32)^2 + 1000  = 471

Rearranging 1000 - 471 = (x-32)^2

If you solve this, you'll get two values of x, both positive.

At those points we have = 471.

To one side of each x the profit will be more; the other side less.  If you go to the MIF function grapher

https://www.mathsisfun.com/data/function-grapher.php

and enter the function -(x - 32)^2 + 1000 you will see a typical quadratic graph rising to a maximum and the dropping away again.

A horizontal line at y =471 will cut the curve at those two values of x, and you can see that the profit is higher than 471 at all x values between them.  So the answer will have the form (lower number) ≤ x ≤ (higher number) where those two numbers are your answers from above.

You can set your answer out like this:

-(x - 32)^2 + 1000  ≥  471  so 1000 - 471 ≥  (x-32)^2

So √529    ≥  x-32  ≥   -√529

Subtract 32 to complete the question.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

jadewest

Member

Registered: 2021-02-20

Posts: 44

Re: Solving Quadratic Inequalities

Hello,

I also have trouble completing this exercise:

-x^2 - 12x - 32 > 0

Thank you so much,
Jade

Bob

Administrator

Registered: 2010-06-20

Posts: 10,140

Re: Solving Quadratic Inequalities

That quadratic will factorise with whole numbers so you won't need the formula

So you'll find two values of x where the quadratic = zero.

Then try a value just to the left and just to the right to find when it's positive and when negative for each x you've found.

Or use the grapher to investigate the graph.

If the -x^2 term is bothering you,  you could times each side by -1 to make it +x^2.  Be careful though .... when you times (or divide) an inequality by anything negative you must also reverse the inequality sign.  So it would become:

+x^2 +12x +32 < 0

Hope that helps,

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob