A shipping service restricts the dimensions of the boxes it will ship for a certain type of service. The restriction states that for boxes shaped like rectangular prisms, the sum of the perimeter of the base of the box and the height of the box cannot exceed 100 inches. The perimeter of the base is determined using the width and length of the box. If a box has a height of 30 inches and its length is 2 times its width, which inequality shows the allowable width \(x\), in inches, of the box?

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Carefully convert each sentence into its corresponding algebraic equation or inequality.

The sum of perimeter of the base of the box and the height of the box cannot exceed 100 inches.

$$ P+h \leq 100 $$

The perimeter of the base is determined using the width and length of the box.

$$ P=2l+2w $$

If a box has a height of 30 inches and its length is 2 times the width

$$ h=30 $$ $$ l=2w $$

Width \(x\)

$$ w=x $$

Substitute accordingly so that we are left with width:

$$ P+h \leq 100 $$ $$ 2l+2w+h \leq 100 $$ $$ 2l+2w+30 \leq 100 $$ $$ 2(2w)+2w+30 \leq 100 $$ $$ 6w \leq 70 $$ $$ 6x \leq 70 $$ $$ x \leq \frac{35}{3} $$

A box cannot have a negative or zero width:

$$ x \gt 0 $$ $$ \boxed {0 \lt x \leq \frac{35}{3}} $$