Given:
A rectangle has a length 5 meters more than five times the width.
The area of the rectangle is less than 100 meters squared.
To find:
The expression or inequality that represents all possible widths of the rectangle.
Solution:
Let x be the width of the rectangle.
Length of the rectangle is 5 meters more than five times the width.
[tex]length=5x+5[/tex]
Area of rectangle is
[tex]Area=length \times width[/tex]
[tex]Area=(5x+5) \times x[/tex]
[tex]Area=5x^2+5x[/tex]
The area of the rectangle is less than 100 meters squared.
[tex]5x^2+5x<100[/tex]
[tex]5x^2+5x-100<0[/tex]
Divide both sides by 5.
[tex]x^2+x-20<0[/tex]
[tex]x^2+5x-4x-20<0[/tex]
[tex]x(x+5)-4(x+5)<0[/tex]
[tex](x+5)(x-4)<0[/tex]
It is true if one factor is negative and other is positive. So,
[tex]x-4<0\Rightarrow x<4[/tex] ...(i)
[tex]x+5>0\Rightarrow x>-5[/tex] ...(ii)
Using (i) and (ii), we get
[tex]-5<x<4[/tex]
Therefore, the required expression or inequality for possible
widths of the rectangle is [tex]-5<x<4[/tex].
