Question

How do you write an inequality and solve given "three fourths of a number decreased by nine is at least forty two"?

Answer 1

The inequality is: `3/4n - 9 >= 42`

The solution is: `n >= 68`

Explanation:

First, let's call "a number" - `n`

"three fourths of a number" then can be written as:

`3/4n`

This, "decreased by nine" then can be written as:

`3/4n - 9`

"is at least" is the same as `>=` so we can now write:

`3/4n - 9 >=`

and what it is "at least" is "forty two" so:

`3/4n - 9 >= 42`

To solve this we first add `color(red)(9)` to each side of the inequality to isolate the `n` term while keeping the inequality balanced:

`3/4n - 9 + color(red)(9) >= 42 + color(red)(9)`

`3/4n - 0 >= 51`

`3/4n >= 51`

Now, we multiply each side of the inequality by `color(red)(4)/color(blue)(3)` to solve for `n` while keeping the inequality balanced:

`color(red)(4)/color(blue)(3) xx 3/4n >= color(red)(4)/color(blue)(3) xx 51`

`cancel(color(red)(4))/cancel(color(blue)(3)) xx color(blue)(cancel(color(black)(3)))/color(red)(cancel(color(black)(4)))n >= color(red)(4) xx 17`

`n >= 68`

Answer 2

Without any punctuation there are different interpretations which lead to different solutions:

`3/4 x -9 >= 42" or "3/4(x-9) >=42`

`x >=56" or "x >=65`

Explanation:

Let's use some punctuation so show exactly what is meant:
note how the placement of a comma gives a different meaning.

`color(blue)("three fourths of a number"), color(red)(" decreased by 9") color(limegreen)(" is at least 42")`

Let the number be `x`

`color(blue)(3/4x) color(red)(-9) color(limegreen)( >=42)`

Solving gives:

`3/4x >= 51`

`4/3 xx 3/4x >= 51 xx 4/3`

`x >=56`

However, if we interpret it differently we could have:

`color(blue)("three fourths of,") color(red)(" a number decreased by 9,") color(limegreen)(" is at least 42")`

`color(blue)(3/4) color(red)((x-9)) color(limegreen)( >= 42`

Solving gives:

`4/3 xx 3/4(x-9) >= 42 xx 4/3`

`x-9 >= 56`

`x >=65`