Question

A rectangle has a length 5 meters more than five times the

width. The area of the rectangle is less than 100 meters
squared. Write an expression that represents all possible
widths (use the variable x) of the rectangle. Write your answer
as an inequality, a

Answer 1

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.

`length=5x+5`

Area of rectangle is

`Area=length * width`

`Area=(5x+5) * x`

`Area=5x^2+5x`

The area of the rectangle is less than 100 meters squared.

`5x^2+5x<100`

`5x^2+5x-100<0`

Divide both sides by 5.

`x^2+x-20<0`

`x^2+5x-4x-20<0`

`x(x+5)-4(x+5)<0`

`(x+5)(x-4)<0`

It is true if one factor is negative and other is positive. So,

`x-4<0\Rightarrow x<4` ...(i)

`x+5>0\Rightarrow x>-5` ...(ii)

Using (i) and (ii), we get

`-5<x<4`

Therefore, the required expression or inequality for possible

widths of the rectangle is
`-5<x<4`.