### Home > AC > Chapter 9 > Lesson 9.2.1 > Problem9-32

9-32.

Thui made the following hypotheses: $2n − 1 \lt 5$ and $n + 1 \lt 2n$. Which of the following conclusions can she make?

1. $1 \le n \le 3$

1. $1 \le n \lt 3$

1. $1 \lt n \le 3$

1. $1 \lt n \lt 3$

Find the boundary points of the two equations.

$2n − 1 = 5 n + 1 = 2n$
$2n = 6 1 = n$
$n = 3$

$\text{Boundary points } = 3 \text{ and } 1$

Test numbers on all sides of the boundary points.

Test $n = 0$
$2(0) −1 \lt 5 0 +1 ≤ 2\left(0\right)$
$−1 \lt 5 1 ≤ 0$
True False

Test $n = 2$
$2\left(2\right) −1 \lt 5 2 +1 ≤ 2\left(2\right)$
$3 \lt 5 3 ≤ 4$
True True

Test $n = 4$
$2\left(4\right) +1 \lt 5 4 + 1 ≤ 2\left(4\right)$
$9 \lt 5 5 ≤ 8$
False True

Because $n = 2$ is the only test that makes both equations true, (b), or $1 ≤ n \lt 3$is the correct answer.