Chapter 3, Problem 1FCCE

State the inequality properties for addition, subtraction, and multiplication.

To determine

The inequality properties for addition, subtraction and multiplication.

Answer to Problem 1FCCE

Solution:

Inequality property for addition: $\text{If}a<b\text{,}\text{then}a+c<b+c$.

Inequality property for subtraction: $\text{If}a<b\text{,}\text{then}a-c<b-c$.

Inequality property for multiplication: $\begin{array}{l}\text{If}a<b\text{,}c>0\text{then}ac<bc\\ \text{If}a<b\text{,}c<0\text{then}ac>bc\end{array}$.

Explanation of Solution

Let $a\text{ and }b$ represent real numbers.

Inequality Property for Addition:

Suppose that $a<b$ and that c is any number.

Then, $a+c<b+c$.

For example, consider the inequality $2<3$ and add $4$ to both sides to get

$2+4<3+4$.

That is, $6<7$, a correct inequality.

Inequality Property for Subtraction:

Suppose that $a<b$ and that c is any number.

Then, $a-c<b-c$.

For example, consider the inequality $2<3$ and subtract $1$ to both sides to get

$2-1<3-1$.

That is, $1<2$, a correct inequality.

Inequality Property for Multiplication:

Suppose that $a<b$ and that c is any positive number.

Then, $ac<bc$.

For example, consider the inequality $2<3$ and multiply $2$ to both sides to get

$2\times 2<3\times 2$.

That is, $4<6$, a correct inequality.

Suppose that $a<b$ and that c is any negative number.

Then, $ac>bc$.

For example, consider the inequality $2<3$ and multiply $\left(-2\right)$ to both sides to get

$2\times \left(-2\right)<3\times \left(-2\right)$.

That is, $-4>-6$, a correct inequality.