Question

How do you solve `\frac { 1+ m } { 10} - \frac { m + 2} { 6} < - \frac { m } { 30}`?

Answer 1

See a solution process below:

Explanation:

First, multiply each side of the inequality by `color(red)(30)` to eliminate the fractions while keeping the inequality balanced. We are using `color(red)(30` because it is the Lowest Common Denominator for the three fractions:

`color(red)(30)((1 + m)/10 - (m + 2)/6) < color(red)(30) xx -m/30`

`(color(red)(30) xx (1 + m)/10) - (color(red)(30) xx (m + 2)/6) < cancel(color(red)(30)) xx -m/color(red)(cancel(color(black)(30)))`

`(cancel(color(red)(30))" " 3 xx (1 + m)/color(red)(cancel(color(black)(10)))) - (cancel(color(red)(30))" "5 xx (m + 2)/color(red)(cancel(color(black)(6)))) < -m`

`3(1 + m) - 5(m + 2) < -m`

`(3 xx 1) + (3 xx m) - (5 xx m) - (5 xx 2) < -m`

`3 + 3m - 5m - 10 < -m`

Next, we can group and combine like terms on the left side of the inequality:

`3 - 10 + 3m - 5m < -m`

`(3 - 10) + (3 - 5)m < -m`

`-7 + (-2)m < -m`

`-7 - 2m < -m`

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

`-7 - 2m + color(red)(2m) < -m + color(red)(2m)`

`-7 - 0 < -1m + color(red)(2m)`

`-7 < (-1 + color(red)(2))m`

`-7 < 1m`

`-7 < m`

We can reverse of "flip" the entire inequality to state the solution in terms of `m`:

`m > -7`