### Home > CCA > Chapter 9 > Lesson 9.3.1 > Problem9-74

9-74.

Thui made the following hypotheses: $2n−1<5$ and $n+1\le2n$. Which of the following conclusions can she make?

1. $n \geq 1 \text { and } n \leq 3$

1. $n \geq 1 \text { and } n < 3$

1. $n > 1 \text { and } n \leq 3$

1. $n > 1 \text { and } n < 3$

Find the boundary points of the two equations.

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

Boundary points $=3$ and $1$

Test numbers on all sides of the boundary points.

Test $n=0$
$2(0)−1<5\ \ \ \ \ \ \ \ 0+1\le2(0)$
$−1<5\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\le0$
True                           False

Test $n=2$
$2(2)−1<5\ \ \ \ \ \ \ \ \ 2+1\le2(2)$
$3<5\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3\le4$
True                             True

Test $n=4$
$2(4)+1<5\ \ \ \ \ \ \ \ \ 4+1\le2(4)$
$9<5\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 5\le8$
False                           True

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