Question

How do you solve and write the following in interval notation: `|6t-6|<6`?

Answer 1

See full solution process below

Explanation:

Because this is a problem contain the absolute value function we must solve the problem for both the negative and positive forms of the problem or in this case +6 and -6.

Also, because it is an inequality we must solve it as a system of inequalities as shown below.

We can rewrite this problem as:

`-6 < 6t - 6 < 6`

We can now solve while ensuring we perform all operations to each portion of the system of inequalities.

First we will add `color(red)(6)` to each portion of the system:

`-6 + color(red)(6) < 6t - 6 + color(red)(6) < 6 + color(red)(6)`

`0 < 6t - 0 < 12`

`0 < 6t < 12`

Next we will divide each portion of the system by `color(red)(6)` to solve for `t` while keeping the system balanced:

`0/color(red)(6) < (6t)/color(red)(6) < 12/color(red)(6)`

`0 < (color(red)(cancel(color(black)(6)))t)/cancel(color(red)(6)) < 2`

`0 < t < 2`

Writing this solution in interval form gives:

(0, 2)