# At a Glance - Inequalities

We have four possible inequality signs to choose from: <, >, ≤, and ≥. There's also the slightly less popular ^, which means that everything below is greater than everything above, but that brings us into a touchy area, spiritually speaking. Translating English into one of these symbols is a matter of common sense. Good thing you have some of that.

The signs < and > are usually straightforward.

- "Three is less than
*x*'' can be abbreviated by 3 <*x*.

- "5 is more than
*x*'' can be written in symbols as 5 >*x*.

When we're translating into statements that use ≤ or ≥, we have a slightly wider range of English phrases to use. For example, each of the following English statements translates into symbols as *x* ≤ 20.

*x*is**less than or equal to**twenty.

*x*is**at most**twenty.

*x*is**not more than**twenty.

*x*is**not greater than**twenty.

- Twenty could be more than
*x*, or it could be exactly as much as*x*. You'll never know. Na-nana-boo-boo.

Each of these following English statements translates into symbols as *x* ≥ 0.

*x*is**greater than or equal to**zero.

*x*is**not less than**zero.

*x*is**not smaller than**zero.

*x*is**at least**zero.

#### Exercise 1

Translate the English sentence into a mathematical equation or inequality as appropriate:

Three *y* plus seven equals forty-two.

#### Exercise 2

Translate the English sentence into a mathematical equation or inequality as appropriate:

Seven-eighths of *x* is at most thirty.

#### Exercise 3

Translate the English sentence into a mathematical equation or inequality as appropriate:

Nine is less than *y*.

#### Exercise 4

Translate the English sentence into a mathematical equation or inequality as appropriate:

Three less than one-fifth of *y* is eleven.

#### Exercise 5

Translate the English sentence into a mathematical equation or inequality as appropriate:

*z* minus one is at least negative two.

#### Exercise 6

Translate the English sentence into a mathematical equation or inequality as appropriate:

If we quadruple *x* and then add four we get eight.

#### Exercise 7

Translate the English sentence into a mathematical equation or inequality as appropriate:

The ratio of *x* and four is greater than the difference of *x* and two.

#### Exercise 8

Translate the English sentence into a mathematical equation or inequality as appropriate:

The total of *x* and five is no more than four.

#### Exercise 9

Translate the English sentence into a mathematical equation or inequality as appropriate:

Three times *z* yields fifty-one.

#### Exercise 10

Translate the English sentence into a mathematical equation or inequality as appropriate:

*x* combined with the difference of *x* and four yields three.