Question

Andrew is creating a rectangular dog run in his back yard. The length of the dog run is 22 feet. The perimeter of the dog run must be at least 48 feet and no more than 88 feet. Use a compound inequality to find the range of values for the width of the dog run. Use x when writing your inequality.

Answer 1

`2\leq x\leq22`

Step-by-step explanation:

Here, we want to get the range of values for the width of the rectangle

We start by representing the width by a variable x

Mathematically, we have the perimeter of the rectangle as:

`2(l\text{ + w)}`

where l is the length and w is the width of the rectangle

The perimeter in terms of the actual values is thus:

`2(22\text{ + x) }`

This value is at least 48 ft:

`2(22+x)\text{ }\ge\text{ 48 or 48 }\leq\text{ 2(22+x)}`

The perimeter is no longer than 88 ft

We have that as:

`2(22+x)\text{ }\leq\text{ 88}`

We have the compound inequality as:

`48\text{ }\leq\text{ 2(22+x) }\leq\text{ 88}`

We then proceed to solve the compound inequality as follows:

`\begin{gathered} 48\text{ }\leq44\text{ + 2x} \\ 48-44\leq\text{ 2x} \\ 4\leq2x \\ 2\leq x \end{gathered}`

Secondly:

`\begin{gathered} 2(22+x)\leq\text{ 88} \\ 44\text{ + 2x}\leq88 \\ 2x\leq88-44 \\ 2x\leq44 \\ x\leq22 \end{gathered}`

Thus, we have the solution as:

`2\leq x\leq22`