Question

How do you solve and write the following in interval notation: `5[5m-(m+8)]> -8(m-6)`?

Answer 1

See the entire solution process below:

Explanation:

First, expand and then group and combine like terms within the brackets [ ]:

`5[5m - (m + 8)] > -8(m - 6)`

`5[5m - m - 8] > -8(m - 6)`

`5[5m - 1m - 8] > -8(m - 6)`

`5[(5 - 1)m - 8] > -8(m - 6)`

`5[4m - 8] > -8(m - 6)`

Next, expand the terms within the brackets and parenthesis by multiplying each term within the brackets/parenthesis by the term outside the brackets/parenthesis:

`color(red)(5)[4m - 8] > color(blue)(-8)(m - 6)`

`(color(red)(5) * 4m) - (color(red)(5) * 8) > (color(blue)(-8) * m) + (color(blue)(-8) * -6)`

`20m - 40 > -8m + 48`

Then add `color(red)(40)` and `color(blue)(8m)` to each side of the inequality to isolate the `m` term while keeping the inequality balanced:

`20m - 40 + color(red)(40) + color(blue)(8m) > -8m + 48 + color(red)(40) + color(blue)(8m)`

`20m + color(blue)(8m) - 40 + color(red)(40) > -8m + color(blue)(8m) + 48 + color(red)(40)`

`(20 + color(blue)(8))m - 0 > 0 + 88`

`28m > 88`

Now, divide each side of the inequality by `color(red)(28)` to solve for `m` while keeping the inequality balanced:

`(28m)/color(red)(28) > 88/color(red)(28)`

`(color(red)(cancel(color(black)(28)))m)/cancel(color(red)(28)) > (4 xx 22)/color(red)(4 xx 7)`

`m > (color(red)(cancel(color(black)(4))) xx 22)/color(red)(color(black)(cancel(color(red)(4))) xx 7)`

`m > 22/7`