2.3: Arithmetic of inequality

Definition: Inequality

Let $a, b\in \mathbb{Z}$. Then

1. $a< b$ provided $b=a + k$, for some $k \in \mathbb{Z_+}$.
2. $a> b$ provided $a=b + h$, for some $h \in \mathbb{Z_+}$.

Theorem $\PageIndex{1}$

Let $a, b\in \mathbb{Z}$.

1. If $a< b$ then $a+c< b+c$, $\forall c \in \mathbb{Z}$.
2. If $a< b$ then $ac< bc$, $\forall c \in \mathbb{Z_+}$.
3. If $a< b$ then $ac> bc$, $\forall c \in \mathbb{Z_-}$.
4. If $a< b$ and $c< d$ then $a+c< b+d$.

Proof

Let $a, b, c \in \mathbb{Z}$ such that $a <b$. Then $b=a + k$, for some $k \in \mathbb{Z_+}$.

1. Now consider, $b+c= (a+k)+c= (a+c)+k$, for some $k \in \mathbb{Z_+}$. Thus $a+c< b+c$.

2.

Example $\PageIndex{1}$:

Determine all integers $m$ that satisfy $-12m \geq 324$.

Solution

Since $-12m \geq 324$, $m \leq -\dfrac{324}{12}=-27$.

Example $\PageIndex{2}$:

Determine all integers $m$ that satisfy $14m \geq 635$.

Solution

Since $14m \geq 635$, $m \geq \dfrac{635}{14}=45.35$. Thus the solutions are $\{m\in \mathbb{Z}| m\geq 46\}.$

Example $\PageIndex{3}$:

Determine all integers $k$ that satisfy $-165+ 98k \geq 0, -335+199k \geq 0, -165+ 98k < 100$ and $-335+199k <100$.

Solution

Since $-165 + 98k ≥ 0, k ≥ 1.68$.

Since $-335 + 199k ≥ 0, k ≥ 1.68$.

Since $-165 + 98k < 100, 98k < 265,$ and $k < 2.70.$

Since $-335 + 199k < 100, 199k < 435,$ and $k < 2.18.$

Since $1.68 ≤ k < 2.18$ and $k ∈ ℤ, k = 2.$