Question

Which number is a solution of the inequality? x(7 – x) > 8

Answer 1

X(7-x)>8  perform indicated multiplication on left side

7x-x^2>8  divide both sides by -1

x^2-7x<-8  halve the linear coefficient, square it, and add it to both side ie (-7/2)^2=12.25

x^2-7x+12.25<4.25 now the left side is a perfect square

(x-3.5)^2<4.25  take the square root of both sides

x-3.5<±√4.25  add 3.5 to both sides

x<3.5±√4.25

So x>3.5-√4.25 and x<3.5+√4.25  or if you want approximations

x>1.44 and x<5.56 or in interval notation:

x=(1.44, 5.56)  (to nearest hundredths)

Answer 2

Expand
7x-x²>8
times -1 both sides and flip sign
x²-7x<-8
addd 8 to both sides
x²-7x+8<0
if we solve assuming x=0 to find where it is equal to 0 we get
x=(7-√17)/2 and x=(7+√17)/2
we want less than 0
test a point betwen them
for x=3, we get
-4<0
true
so we want the numbers between x=(7-√17)/2 and x=(7+√17)/2

the number(s) we want is(are) between (7-√17)/2 and (7+√17)/2