Question

How do you solve `\frac { 4} { 3} m - 3< m + \frac { 2} { 3} - \frac { 1} { 3} m`?

Answer 1

See the entire solution process below:

Explanation:

First, I would multiply each side of the inequality by `color(red)(3)` to eliminate the fractions while keeping the inequality balanced:

`color(red)(3)(4/3m - 3) < color(red)(3)(m + 2/3 - 1/3m)`

`(color(red)(3) * 4/3m) - (color(red)(3) * 3) < (color(red)(3) * m) + (color(red)(3) * 2/3) - (color(red)(3) * 1/3m)`

`(cancel(color(red)(3)) * 4/color(red)(cancel(color(black)(3)))m) - 9 < 3m + (cancel(color(red)(3)) * 2/color(red)(cancel(color(black)(3)))) - (cancel(color(red)(3)) * 1/color(red)(cancel(color(black)(3)))m)`

`4m - 9 < 3m + 2 - 1m`

Next, group and combine like terms on the right side of the inequality:

`4m - 9 < 3m - 1m + 2`

`4m - 9 < (3 - 1)m + 2`

`4m - 9 < 2m + 2`

Then, add `color(red)(9)` and subtract `color(blue)(2m)` from each side of the inequality to isolate the `m` term while keeping the inequality balanced:

`4m - 9 + color(red)(9) - color(blue)(2m) < 2m + 2 + color(red)(9) - color(blue)(2m)`

`4m - color(blue)(2m) - 9 + color(red)(9) < 2m - color(blue)(2m) + 2 + color(red)(9)`

`(4 - color(blue)(2))m - 0 < 0 + 11`

`2m < 11`

Now, divide each side of the inequality by `color(red)(2)` to solve for `m` while keeping the inequality balanced:

`(2m)/color(red)(2) < 11/color(red)(2)`

`(color(red)(cancel(color(black)(2)))m)/cancel(color(red)(2)) < 11/2`

`m < 11/2`