# How do you write the inequality and solve "the sum of 3 times a number and 4 is between -8 and 10#?

Jan 16, 2017

$- 8 < 3 n + 4 < 10 \iff - 4 < n < 2$

#### Explanation:

First off we have the statement "3 times a number plus 4".
Part of the fun of math is variables. It can be any symbol you want to stand for the number. For simplicity I chose the letter n for "number".

So the English statement "3 times a number plus 4" becomes $3 n + 4$.

Since $3 n + 4$ is "in between -8 and 10" we know that -8 is less than $3 n + 4$ and 10 is greater.

So,

$- 8 < 3 n + 4$

and

$3 n + 4 < 10$

these two statements can be combined into

$- 8 < 3 n + 4 < 10$

Now it's time to solve.

$- 8 < 3 n + 4 < 10$

$\iff$

$- 8 - 4 < 3 n + 4 - 4 < 10 - 4$ subtract 4 from each part

$\iff$

$- 12 < 3 n < 6$

Now we can divide by 3, because it's a common factor

$\iff$

$- 4 < n < 2$

So now we know that n is between -4 and 2. The inequality is simplified.