Elouise is creating a rectangular garden in her backyard. The length of the garden is 12 feet. The perimeter of the garden must be at least 36 feet and no more than 48 feet. Use a compound inequality to find the range of values for the width of the garden.

The compound inequality used to define the range of values of the width of the garden w is 6 ≤ w ≤ 12. This gives us the range of values for the width of the garden as {6 feet, 7 feet, 8 feet, 9 feet, 10 feet, 11 feet, 12 feet}.

What do we mean by compound inequality?

Compound equality is a statement including more than one inequality operation (>, <, ≥, ≤, ≠).

How do we solve the given question?

In the question, we are informed that Elouise is creating a rectangular garden in her backyard. The length of the garden is 12 feet. The perimeter of the garden must be at least 36 feet and no more than 48 feet.

We are asked to use a compound inequality to find the range of values for the width of the garden.

The perimeter of a rectangle is given by the formula,

Perimeter = 2*(Length + Width)

Let the width of the garden be w feet.

∴ The perimeter of the garden = 2*(Length + Width) = 2*(12 + w).

We have been informed that the perimeter is at least 36 feet. This can be represented by the inequality, 2*(12 + w) ≥ 36

or, 12 + w ≥ 36/2

or, 12 + w ≥ 18

or, w ≥ 18 - 12

or, w ≥ 6 ... (i)

We have been informed that the perimeter is no more than 48 feet. This can be represented by the inequality, 2*(12 + w) ≤ 48

or, 12 + w ≤ 48/2

or, 12 + w ≤ 24

or, w ≤ 24 - 12

or, w ≤ 12 ... (ii).

Combining (i) and (ii), we get the required compound inequality,

6 ≤ w ≤ 12.

This gives us the range of w to be {6 feet, 7 feet, 8 feet, 9 feet, 10 feet, 11 feet, 12 feet}.

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