Question

Translate the sentence into an inequality?

Answer 1

`2c - 6 <= -25`

Explanation:

Since we are using the variable `c`, we have to determine what `c`'s relationship to the other numbers is.

Since the question says twice the difference of a number and 6, and `c` is "a number", we know that we should use `2c - 6` on one side of the inequality. The other side should be "at least -25", which means -25 is on the right, and the inequality should be -25 or greater. Another way to put it is that the `2c - 6` should be less or equal to -25.

Now if we make this an inequality, it becomes:

`2c - 6 <= -25`

If you want to solve:

`2c - 6 <= -25`
`2c - 6 +6 <= -25+6`
`2c <= -19`
`c <= -9.5`

Answer 2

`2abs(c-6) >= -25`

Explanation:

Twice `color(blue)("the difference of a number and 6")` is at least `-25`
`rarr` Twice `color(blue)(abs(c-6))` is at least `-25`

`color(magenta)("Twice ")color(blue)(abs(c-6))` is at least `-25`
`rarr color(magenta)2color(blue)(abs(c-6))` is at least `-25`

`color(magenta)2color(blue)(abs(c-6))color(orange)(" is at least ") -25`
`rarr color(magenta)2color(blue)(abs(c-6))color(orange)(>=)-25`

Notice that this will be true for all Real values of `c`,
so it might be that `color(blue)("the difference of a number and 6")` was intended to be translated as either `color(blue)(""(c-6))` or as `color(blue)(""(6-c))`