Solved Examples on Polynomial Inequalities

$ -8x^2 + 800x \ge 0 \\[3ex] -8(x^2 - 100x) \ge 0 \\[3ex] x^2 - 100x \le \dfrac{0}{-8} \\[5ex] x^2 - 100x \le 0 \\[3ex] x(x - 100) \le 0 \\[3ex] Assume:\;\;x(x - 100) = 0 \\[3ex] x = 0 \;\;\;OR\;\;\; x - 100 = 0 \\[3ex] x = 0 \;\;\;OR\;\;\; x = 100 \\[3ex] Test\;\;Intervals\;\;are: \\[3ex] x \lt 0 \\[3ex] 0 \le x \le 100 \\[3ex] x \gt 100 \\[3ex] $

Let:	x < 0 x = −1	0 ≤ x ≤ 100 x = 1	x > 0 x = 101
$x$	−	+	+
$x - 100$	−	−	+
$x(x - 100)$	+	−	+

Less than or equal to zero means nonnegative
Hence, the second test interval gives the solution of the inequality
Set Notation: {x | 0 ≤ x ≤ 100}
Interval Notation: [0, 100]

Check

[0, 100]

LHS	RHS
$ -8x^2 + 800x \\[3ex] x = 3 \\[3ex] -8(3^2) + 800(3) \\[3ex] -72 + 2400 \\[3ex] 2328 $	0
2328 ≥ 0

$ -8p^2 + 800p \ge 10752 \\[3ex] -8p^2 + 800p - 10752 \ge 0 \\[3ex] -8(p^2 - 100p + 1344) \ge 0 \\[3ex] p^2 - 100p + 1344 \le \dfrac{0}{-8} \\[5ex] p^2 - 100p + 1344 \le 0 \\[3ex] (p - 16)(p - 84) \le 0 \\[3ex] Assume:\;\;(p - 16)(p - 84) = 0 \\[3ex] p - 16 = 0 \;\;\;OR\;\;\; p - 84 = 0 \\[3ex] p = 16 \;\;\;OR\;\;\; p = 84 \\[3ex] Test\;\;Intervals\;\;are: \\[3ex] p \lt 16 \\[3ex] 16 \le p \le 84 \\[3ex] p \gt 84 \\[3ex] $

Let:	p < 16 p = 0	16 ≤ p ≤ 84 p = 17	p > 84 p = 85
$p - 16$	−	+	+
$p - 84$	−	−	+
$(p - 16)(p - 84)$	+	−	+

Less than or equal to zero means nonnegative
Hence, the second test interval gives the solution of the inequality
Set Notation: {p | 16 ≤ p ≤ 84}
Interval Notation: [16, 84]

Check

[16, 84]

LHS	RHS
$ -8p^2 + 800p \\[3ex] p = 20 \\[3ex] -8(20^2) + 800(20) \\[3ex] -8(400) + 16000 \\[3ex] -3200 + 16000 \\[3ex] 12,800 $	10752
12800 ≥ 10752