
# 2.3: Arithmetic of inequality

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Definition

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 $$k ∈ ℤ, k = 2.$$