Chapter 3.4, Problem 1E

To determine

To explain: equivalent inequalities.

Answer to Problem 1E

Equivalent inequalities are inequalities that have the same solution.

Explanation of Solution

Equivalent inequalities are inequalities that have the same solution.

Adding or subtracting the same number on each side of an inequality produces an equivalent inequality.

  $\text{if }a<b,\text{ then }a+c<b+c\text{ and }a-c<b-c.$

In the above statement, adding or subtracting $c$ from $a$ and $b$ respectively,

Left hand side of the equation will always be less than the right hand side.

Similarly,

  $\text{if }a>b,\text{ then }a+c>b+c\text{ and }a-c>b-c.$

Adding or subtracting $c$ from $a$ and $b$ respectively,Left hand side of the equation will always be greater than the right hand side.

Thus,

  $a+c<b+c$ Or $a-c<b-c$ and $a<b$ are equivalent inequalities.

Similarly,

  $a+c>b+c$ Or $a-c>b-c$ and $a>b$ are equivalent inequalities.