Question

PLEASE HELP ME!!!!

Solve: √(x-5)<7

I believe it's either: x<54 or 5≤x<54

Answer 1

First we need to find the domain

Answer 2

since you cannot take the square root of a negative number, this means x-5 can't be negative, so this means x - 5 >= 0

solve for x

x - 5 >= 0

x-5+5 >= 0+5

x >= 5

Answer 3

now let's solve for x in sqrt(x-5) < 7


sqrt(x-5) < 7

( sqrt(x-5) )^2 < ( 7 )^2 ... square both sides

| x - 5 | < 49

-49 < x - 5 < 49

-49+5 < x < 49+5

-44 < x < 54

5 <= x < 54 ... applying the domain restriction here

Answer 4

in line 3, I used the idea that

( sqrt(x) )^2 = |x|



and in line 4, I used the property that

if |x| < k and k is some positive number, then -k < x < k

Answer 5

okay thank you so much, so in final 5≤x<54, but if x is less than 5 the solution is not real but it would still work?

Answer 6

if x was less than 5, then x-5 would be negative (try x = 4 and you'll get x-5 = 4-5 = -1)

but again you cannot take the square root of a negative number

Answer 7

that's why the domain of f(x) = sqrt(x-5) is x>= 5

Answer 8

so it has to be greater than or equal to 5 for the solution to be a real number and less than 54 for the solution to be less than 7,

so you can't include imaginary numbers in inequalities?

Answer 9

you can't include imaginary numbers in inequalities because you can't naturally order imaginary or complex numbers

ex: which number is bigger, 2+i or 3-i ?

there's no way of sorting the two like you can sort real numbers

Answer 10

so inequalities only apply to real numbers

Answer 11

okay thank you so much!!!

Answer 12

you're welcome