Answer:
All numbers not decimals are natural numbers
Step-by-step explanation:
![\bf \sqrt{n}\leqslant \sqrt{4n-6}](https://tex.z-dn.net/?f=%5Cbf%20%5Csqrt%7Bn%7D%5Cleqslant%20%5Csqrt%7B4n-6%7D%3C%5Csqrt%7B2n%2B5%7D%5Cimplies%20%5Cbegin%7Bcases%7D%20%5Csqrt%7Bn%7D%5Cleqslant%20%5Csqrt%7B4n-6%7D%5C%5C%5B1em%5D%20%5Csqrt%7Bn%7D%3C%20%5Csqrt%7B2n%2B5%7D%20%5Cend%7Bcases%7D%20%5C%5C%5C%5C%5B-0.35em%5D%20%5Crule%7B34em%7D%7B0.25pt%7D%5C%5C%5C%5C%20%5Csqrt%7Bn%7D%5Cleqslant%20%5Csqrt%7B4n-6%7D%5Cimplies%20%5Cstackrel%7B%5Ctextit%7Bsquaring%20both%20sides%7D%7D%7Bn%5Cleqslant%204n-6%7D%5Cimplies%200%5Cleqslant%204n-n-6%5Cimplies%200%5Cleqslant%203n-6%20%5C%5C%5C%5C%5C%5C%206%5Cleqslant%203n%5Cimplies%20%5Ccfrac%7B6%7D%7B3%7D%5Cleqslant%20n%5Cimplies%20%5Cboxed%7B2%5Cleqslant%20n%7D%20%5C%5C%5C%5C%5B-0.35em%5D%20%5Crule%7B34em%7D%7B0.25pt%7D)
![\bf \sqrt{n}< \sqrt{2n+5}\implies \stackrel{\textit{squaring both sides}}{n< 2n+5}\implies 0\leqslant 2n - n + 5 \\\\\\ 0 < n+5\implies \boxed{-5 < n} \\\\\\ \stackrel{-5\leqslant n < 2}{\boxed{-5}\rule[0.35em]{10em}{0.25pt}0\rule[0.35em]{3em}{0.25pt}2}](https://tex.z-dn.net/?f=%5Cbf%20%5Csqrt%7Bn%7D%3C%20%5Csqrt%7B2n%2B5%7D%5Cimplies%20%5Cstackrel%7B%5Ctextit%7Bsquaring%20both%20sides%7D%7D%7Bn%3C%202n%2B5%7D%5Cimplies%200%5Cleqslant%202n%20-%20n%20%2B%205%20%5C%5C%5C%5C%5C%5C%200%20%3C%20n%2B5%5Cimplies%20%5Cboxed%7B-5%20%3C%20n%7D%20%5C%5C%5C%5C%5C%5C%20%5Cstackrel%7B-5%5Cleqslant%20n%20%3C%202%7D%7B%5Cboxed%7B-5%7D%5Crule%5B0.35em%5D%7B10em%7D%7B0.25pt%7D0%5Crule%5B0.35em%5D%7B3em%7D%7B0.25pt%7D2%7D)
namely, -5, -4, -3, -2, -1, 0, 1. Excluding "2" because n < 2.
Answer:
z = 3
Step-by-step explanation:
Since the points are collinear then the slopes between the points are equal.
Calculate the slope m using the slope formula
m = ![\frac{y_{2}-y_{1} }{x_{2}-x_{1} }](https://tex.z-dn.net/?f=%5Cfrac%7By_%7B2%7D-y_%7B1%7D%20%20%7D%7Bx_%7B2%7D-x_%7B1%7D%20%20%7D)
with (x₁, y₁ ) = P (2, - 3) and (x₂, y₂ ) = Q (3, - 2)
m =
= 1
Repeat with
(x₁, y₁ ) = Q (3, - 2) and (x₂, y₂ ) = R (8, z )
m =
=
, then
= 1 ( multiply both sides by 5 )
z + 2 = 5 ( subtract 2 from both sides )
z = 3