How do you solve and write the following in interval notation: 7 ≤ x + 4 or x - 1 ≥ 5?

Apr 25, 2017

First, solve each inequality separately:

$7 \le x + 4$
$\therefore 7 - 4 \le x + 4 - 4$
$\therefore 3 \le x$
$\therefore x \ge 3$
$x$ can be any value between $3$ (including) and $\infty$. Therefore, $x \in \left[3 , \infty\right)$. Brackets mean that it is included, while parentheses mean that it is not included. $\infty$ is never included for interval notations.

$x - 1 \ge 5$
$\therefore x - 1 + 1 \ge 5 + 1$
$\therefore x \ge 6$
$x$ can be any value between $6$ (including) and $\infty$. Therefore, $x \in \left[6 , \infty\right)$.

We can combine these two to get $x \in \left[3 , \infty\right) \cap \left[6 , \infty\right)$. This is equivalent to $x \in \left[6 , \infty\right)$.