Chapter A.1, Problem 151AYU

To determine

To find: Write a brief paragraph that illustrates the similarities and differences between “less than” $(<)$ and “less than or equal to” $(\le )$.

Answer to Problem 151AYU

The notation $a<b$ means that $a$ is less than $b$.

The notation $a<b$ means that $a$ is less than or equal to $b$ (or, equivalently, not greater than $b$, or at most $b$); "not greater than" can also be represented by the symbol for "greater than" bisected by a vertical line, "not."

If Sam tried really hard he might be able to cut the rope EXACTLY in half, so each half is 5m, but we know he didn't because we said there was a "shorter" and "longer" length, so we also know:

$S<5$ and $L>5$.

We can put that into our very neat statement here:

$0<S<5<L<10$

And IF we thought the two lengths MIGHT be exactly 5 we could change that to,

$0<S\le 5\le L<10$

Explanation of Solution

The notation $a<b$ means that $a$ is less than $b$.

The notation $a\le b$ means that $a$ is less than or equal to $b$ (or, equivalently, not greater than $b$, or at most $b$); "not greater than" can also be represented by the symbol for "greater than" bisected by a vertical line, "not."

If Sam tried really hard he might be able to cut the rope EXACTLY in half, so each half is 5m, but we know he didn't because we said there was a "shorter" and "longer" length, so we also know:

$S<5$ and $L>5$.

We can put that into our very neat statement here:

$0<S<5<L<10$

And IF we thought the two lengths MIGHT be exactly 5 we could change that to,

$0<S\le 5\le L<10$