Suppose that C=5x^2-40x+98, x≥0 represents the cost, in hundreds of dollars, to produce x thousands of pens. Find the number of pens which can be produced for no more than $2300.

Question

Answer: between ______ thousand and ______ thousand pens.

Answer 1

$2300 is 23 hundreds so:

5x² - 40x + 98 ≤ 23

Make it an equality and solve the equation

5x² - 40x + 75 = 0

x² - 8x + 15 = 0

(x - 3)(x - 5) = 0

x = 3 and x = 5

You now know that x = 3 and x = 5 are boundaries for the inequality but you are not 100% sure which sections make the inequality true (is it less than 3? is it between 3 and 5? Or is it greater than 5?). To find out, create a number line to double check which section of the x-values:

Try a value less than 3, how about zero:

5x² - 40x + 98 ≤ 23

5(0)² - 40(0) + 98 ≤ 23

98 ≤ 23

This is NOT TRUE so values below 3 are NOT solutions to the inequality.

Try a value between 3 and 5, how about 4:

5x² - 40x + 98 ≤ 23

5(4)² - 40(4) + 98 ≤ 23

80 - 160 + 98 ≤ 23

18 ≤ 23

This is TRUE so values between 3 and 5 ARE solutions to the inequality.

Try a value greater than 5, how about 6:

5x² - 40x + 98 ≤ 23

5(6)² - 40(6) + 98 ≤ 23

180 - 240 + 98 ≤ 23

38 ≤ 23

This is NOT TRUE so values above 5 are NOT solutions to the inequality.

So 3 ≤ x ≤ 5 (between 3 and 5 thousand)