Question

The ideal width of a safety belt strap for a certain automobile is 6 cm. The actual width can vary by at most 0.45 cm. Write an absolute value inequality for the range of acceptable widths and solve the inequality.

Answer 1

`|x-6|\le 0.45`

`x\in [5.55,6.45]`

Explanation:

Absolute Value Inequality

Assume the actual width of a safety belt strap for a certain automobile is x. We know the ideal width of the strap is 6 cm. This means the variation from the ideal width is x-6.

Note if x is less than 6, then the variation is negative. We usually don't care about the sign of the variation, just the number. That is why we need to use the absolute value function.

The variation (unsigned) from the ideal width is:

`|x-6|`

The question requires that the variation is at most 0.45 cm. That poses the inequality:

`|x-6|\le 0.45`

That is the range of acceptable widths. Let's now solve the inequality.

To solve an inequality for an absolute value less than a positive number N, we write:

`-0.45\le x-6 \le 0.45`

This is a double inequality than can be easily solved by adding 6 to all the sides.

`-0.45+6\le x \le 0.45+6`

Operating:

`5.55\le x \le 6.45`

That is the solution in inequality form. Expressing in interval form:

`\boxed{x\in [5.55,6.45]}`