We know there are many operation symbols that are used in algebra. Now, we’ll translate word phrases into algebraic expressions and equations. The symbols and variables we’ve talked about will help us do that. They are summarized below.

Look closely at these phrases using the four operations:

• the sum of [latex]a[/latex] and [latex]b[/latex]
• the difference of [latex]a[/latex] and [latex]b[/latex]
• the product of [latex]a[/latex] and [latex]b[/latex]
• the quotient of [latex]a[/latex] and [latex]b[/latex]

Each phrase tells you to operate on two numbers. Look for the words of and and to find the numbers.

Watch the video below to better understand how to write algebraic expressions from statements.

We’ll eventually apply our skills in algebra to solving equations in complex word problems. Usually start by translating a word phrase to an algebraic equation. Remember, an equation has an equal sign between two algebraic expressions. So if we have a sentence that tells us that two phrases are equal, we can translate it into an equation. We look for clue words that mean equals. Some words that translate to the equal sign are:

• is equal to
• is the same as
• is
• gives
• was
• will be

It may be helpful to put a box around the equals word(s) in the sentence to help you focus separately on each phrase. Then translate each phrase into an expression, and write them on each side of the equal sign.

example

Translate the sentence into an algebraic equation: The sum of [latex]6[/latex] and [latex]9[/latex] is [latex]15[/latex].

Solution
The word is tells us the equal sign goes between 9 and 15.

Locate the “equals” word(s).
Write the = sign.
Translate the words to the left of the equals word into an algebraic expression.
Translate the words to the right of the equals word into an algebraic expression.

example

Translate the sentence into an algebraic equation: Twice the difference of [latex]x[/latex] and [latex]3[/latex] gives [latex]18[/latex].

Show Answer

Solution

Locate the “equals” word(s).
Recognize the key words: twice; difference of …. and ….	Twice means two times.
Translate.
	[latex]2(x-3)=18[/latex]

Now let’s apply our understanding of translating words to algebra in a real world scenario.

In the following video we show more examples of how to write basic algebraic expressions from words, and simplify.

Let’s practice translating sentences into algebraic equations and then solving them.

In the following video we show more examples of how to translate an equation into words and solve. Note that this is different from the written examples on this page because we start with the mathematical equation then translate it into words.

What if we are working with expressions that are not equal? An inequality is used in algebra to compare two quantities that may have different values. The number line can help you understand inequalities. Remember that on the number line the numbers get larger as they go from left to right. So if we know that [latex]b[/latex] is greater than [latex]a[/latex], it means that [latex]b[/latex] is to the right of [latex]a[/latex] on the number line. We use the symbols [latex]\text{<}[/latex] and [latex]\text{>}[/latex] for inequalities.

[latex]a<b[/latex] is read [latex]a[/latex] is less than [latex]b[/latex]
[latex]a[/latex] is to the left of [latex]b[/latex] on the number line

[latex]a>b[/latex] is read [latex]a[/latex] is greater than [latex]b[/latex]
[latex]a[/latex] is to the right of [latex]b[/latex] on the number line

The expressions [latex]a<b\text{ and }a>b[/latex] can be read from left-to-right or right-to-left, though in English we usually read from left-to-right. In general,

[latex]\begin{array}{l}a<b\text{ is equivalent to }b>a.\text{ For example, }7<11\text{ is equivalent to }11>7.\hfill \\ a>b\text{ is equivalent to }b<a.\text{ For example, }17>4\text{ is equivalent to }4<17.\hfill \end{array}[/latex]

When we write an inequality symbol with a line under it, such as [latex]a\le b[/latex], it means [latex]a<b[/latex] or [latex]a=b[/latex]. We read this [latex]a[/latex] is less than or equal to [latex]b[/latex]. Also, if we put a slash through an equal sign, [latex]\ne[/latex], it means not equal.

We summarize the symbols of equality and inequality in the table below.

In the following video we show more examples of how to write inequalities as words.

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