Question

How do you solve `\frac { x + 3} { 9} > \frac { x - 1} { 3}`?

Answer 1

See a solution process below:

Explanation:

First, multiply each side of the inequality by `color(red)(9)` to eliminate the fractions while keeping the inequality balanced:

`color(red)(9) xx (x + 3)/9 > color(red)(9) xx (x - 1)/3`

`cancel(color(red)(9)) xx (x + 3)/color(red)(cancel(color(black)(9))) > cancel(color(red)(9))3 xx (x - 1)/color(red)(cancel(color(black)(3)))`

`x + 3 > 3(x - 1)`

`x + 3 > (3 xx x) - (3 xx 1)`

`x + 3 > 3x - 3`

Next, subtract `color(red)(x)` and add `color(blue)(3)` to each side of the inequality to isolate the `x` term while keeping the inequality balanced:

`-color(red)(x) + x + 3 + color(blue)(3) > -color(red)(x) + 3x - 3 + color(blue)(3)`

`0 + 6 > -color(red)(1x) + 3x - 0`

`6 > (-color(red)(1) + 3)x`

`6 > 2x`

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

`6/color(red)(2) > (2x)/color(red)(2)`

`3 > (color(red)(cancel(color(black)(2)))x)/cancel(color(red)(2))`

`3 > x`

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

`x < 3`