Question

How do you solve `\frac { - 2x + 9} { 18} \leq \frac { 3x - 12} { 18}`?

Answer 1

See a solution process below:

Explanation:

First, multiply each side of the inequality by `color(red)(18)` to eliminate the fractions while keeping the inequality balanced:

`color(red)(18) xx (-2x + 9)/18 <= color(red)(18) xx (3x - 12)/18`

`cancel(color(red)(18)) xx (-2x + 9)/color(red)(cancel(color(black)(18))) <= cancel(color(red)(18)) xx (3x - 12)/color(red)(cancel(color(black)(18)))`

`-2x + 9 <= 3x - 12`

Next, add `color(red)(2x)` and `color(blue)(12)` to each side of the inequality to isolate the `x` term while keeping the inequality balanced:

`color(red)(2x) - 2x + 9 + color(blue)(12) <= color(red)(2x) + 3x - 12 + color(blue)(12)`

`0 + 21 <= (color(red)(2) + 3)x - 0`

`21 <= 5x`

Now, divide each side of the inequality by `color(red)(5)` to solve for `x` while keeping the inequality balanced:

`21/color(red)(5) <= (5x)/color(red)(5)`

`21/5 <= (color(red)(cancel(color(black)(5)))x)/cancel(color(red)(5))`

`21/5 <= x`

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

`x >= 21/5`