Question

How do you solve and graph `-n+6<7n+4`?

Answer 1

See a solution process below:

Explanation:

First, add `color(red)(n)` and subtract `color(blue)(4)` from each side of the inequality to isolate the `n` term while keeping the inequality balanced:

`color(red)(n) - n + 6 - color(blue)(4) < color(red)(n) + 7n + 4 - color(blue)(4)`

`0 + 2 < color(red)(1n) + 7n + 0`

`2 < (color(red)(1) + 7)n`

`2 < 8n`

Now, divide each side of the inequality by `color(red)(8)` to solve for `n` while keeping the inequality balanced:

`2/color(red)(8) < (8n)/color(red)(8)`

`1/4 < (color(red)(cancel(color(black)(8)))n)/cancel(color(red)(8))`

`1/4 < n`

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

`n > 1/4`

To graph this we will draw a vertical line at `1/4` on the horizontal axis.

The line will be a dashed line because the inequality operator does not contain an "or equal to" clause.

We will shade to the right side of the line because the inequality operator also contains a "greater than" clause:

graph{x> 1/4 [-2, 2, -1, 1]}