Negative three times a number plus four is no more than the number minus eight. What is the number?

Let the number be `n`

Three times the number is `-3 xx n` = `3n`

Plus four is `-3n +4`

Number minus eight is `n - 8`

No more than is `<=`

So we get:

`-3n+4 <= n - 8`

Simplify and solve this linear equation:

`-3n-n <= -8-4`

`-4n <= -12`

`n >= -12/-4`

`n >= 3`

So the number `n` is such that `n>=3`

Hope this helps!

"a number" `-> n`

"Negative three times a number" `-> -3n`

"Negative three times a number plus four" `-> -3n + 4`

"is no more than" `-> <=` giving:

`-3n <=`

"the number minus eight" `-> n - 8` giving:

`-3n + 4 <= n - 8`

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

`-3n - color(red)(n) + 4 - color(blue)(4) <= n - color(red)(n) - 8 - color(blue)(4)`

`-3n - color(red)(1n) + 0 <= 0 - 12`

`(-3 - color(red)(1))n <= -12`

`-4n <= -12`

Now, divide each side of the inequality by `color(blue)(-4)` to solve for `n` while keeping the inequality balanced. However, because we are multiplying or dividing an inequality by a negative number we must reverse the inequality operator.

`(-4n)/color(blue)(-4) color(red)(>=) (-12)/color(blue)(-4)`

`(color(blue)(cancel(color(black)(-4)))n)/cancel(color(blue)(-4)) color(red)(>=) 3`

`n >= 3`