Creating Expressions, Equations, and Inequalities
One of the most critical skills in algebra is learning to write an equation. Being able to translate a problem -- whether a word problem or a real life problem -- into an equation opens up an entire realm of math.
So, how do you translate words into an equation?
First, some basics:
- Expressions, equations, and inequalities contain numbers. If you are given a relevant number, figure out how it fits into the equation.
- Numbers are often unknown. These unknown numbers should be represented by variables. Variables can be any letter. We often use x and n, but any letters will do. Generally, stay away from variables that look like numbers (e.g., l and g). If you are using several variables, it's good to have letters that are distinct (c and r can get confusing). If you are doing a word problem, it sometimes helps to use letters that represent items (e.g., if the farmer has more wheat then corn, then $w>c$ makes sense).
- Several words signal that you need to insert a variable:
| Key Words | What number... | A number... | What quantity | What amount |
| Examples: | What number is equal to 3x and 4x-1?
| A number is twice as big as 3...
| What quantity of apples is greater than 3 but less than 7?
| What amount of glue is need to fill 3 4-quart containers?
|
- An equation contains an equal sign $=$
- An inequality is like an equation, but it contains a greater than $>$, less than $<$, greater or equal to $\geq$, or less than or equal to $\leq$ sign.
- An expression is set of numbers, variables, and operations but does not have an equal or inequality sign. Expressions can be simplified, but they cannot be solved.
| Symbol | Equal $=$ | Greater than $>$ | Less than $<$ | Greater than or equal to $\geq$ | Less than or equal to $\leq$ |
| Key Words | equals | is greater than is more than is larger than >exceeds | is less than is lower than is smaller than | is greater than or equal to is at least is no less than | is less than or equal to |
| Examples | A number is the same as 6: $n\mathbf{ = } 6$ | x is more than y: $x\mathbf{>}3+y$ | g is less then 10: $g\mathbf{<}10$ | x is no less than 64: $x\mathbf{\geq}64$ | He earns no more than \$20/hour $m\mathbf{\leq}20$ |
And finally, what really defines an expression/equation/inequality are the operations. Algebra operations are the same operations you're used to using in math every day: addition $+$, subtraction $-$, multiplication $\times$, and division $\div$.
| Symbol | Addition $+$ | Subtraction $-$ | Multiplication $\times$ | Division $\div$ |
| Key Words | add | subtract minus less take away difference change left over decreased by reduced by | multiplied by
| divided by quotient per each |
| Examples | A number plus 6 is 8 $n+6=8$
| The difference between 5 and x is 3 $5-x=3$ 7 is 12 less than x $x-12=7$ | The product of 5 and x is 20 $5\times x=20$
| A number divided into 6 equal groups is 2 $x\div6=2$ |
| Special notes: | You can add numbers in any order (because of the communicative property of addition). | Order matters! It's often switched. So, x less 7 is $x-7$ but 7 less than x is also $x-7$. Think about it: You have \$10. Your friend has three dollars less than you, would you do $3-10$ or $10-3$? Always think about order when you write a subtraction problem. | You can multiple numbers in any order (because of the communicative property of multiplication). "Of" is often used when multiplying by decimals or fractions: one half of 20is ten or 45% of 100 is 45. In both of these cases "of" means multiply. | When you write a division problem either with a division sign ($19\div4$) you enter numbers into a calculator in that order. When you write a division problem as a fraction ($\dfrac{19}{3}$), you enter those numbers into a calculator from top to bottom ($19\div3$). But, if you divide on scratch paper, using the "house," you write numbers in the opposite order: $3\vert\overline{19}$. |
Once you have your numbers, variables, operators, and equal or inequality signs, you have everything you need to put together an expression/equation/inequality (or several!).
Remember, you can piece together an equation (or set of equations) even if you're not entirely sure what you're doing. One of the reasons we write math equations is to make problems clearer. Writing math in words can be clumsy and imprecise. So, let's use key words above to piece together the following equations:
Example: Write an equation that shows the relationship: 7 less than x is equal to the product of x and 2.
Take it step by step:
$$\eqalign{\text{7 less than x }\rightarrow x-7&\text{}&&\mathbf{ Less\; than}\text{ are subtraction words (remember, it's one of the subtraction} \\&\text{}&&\text{words where you have to think about order).}\\\text{Next step: is equal to }\rightarrow x-7&=&&\mathbf{Is \;equal\; to}\text{ means equal sign.}\\\text{Final step: product of x and 2 }\rightarrow x-7&=x\times2&&\mathbf{Product}\text{ is a multiplication word and order does not matter}\\&\text{}&&\text{ in multiplication.}\\x-7&=2x&&\text{Usually in math we show multiplication just by putting numbers}\\&\text{}&&\text{and/or variables attached to each other.}}$$
The one other complicating factor when writing equations is order of operations. If you need to review order of operations see the lesson Order of Operations. Equations follow standard math order of operations. The only way to change that is to use parentheses or brackets. Pay attention when a word problem seems to be grouping numbers or variables.
Example: Fifteen divided by 4 is greater than twice the sum of 13 and 4.
Take it step by step:
$$\eqalign{\text{15 divided by 4 }\rightarrow 15\div 4&\text{}&&\text{}\mathbf{ Divided\; by}\text{ are division words}\\\text{Next step: is greater than }\rightarrow 15\div 4&>&&\text{}\mathbf{ Is\; greater\; than}\text{ means an inequality with the "mouth"} \\&\text{}&&\text{of the inequality sign facing the greater side.}\\\text{Final step: twice the sum of 13 and 5 }\rightarrow 15\div 4& > 2\times &&\text{}\mathbf{ Twice}\text{ is a multiplication word that means multiply by 2.}\\15\div 4&> 2\times (13+4)\text{}&&\mathbf{ Sum}\text{ is an addition word. The way that this is written suggests that }\\\text{}&\text{}&&\text{ it's the sum that isbeing multiplied, not just 13 or 4,}\\&\text{}&&\text{ so group with parentheses.}}$$
Overall, writing equations takes practice (see the lesson on Creating Equations from Word Problems to take this to the next level). But once you memorize the key words and get familiar with the little tricks (like "less than" usually requiring you to switch the order of the numbers), you can write almost any kind of equation, including equations that are embedded in word problems.
Practice Problems:
Reading and Writing Expressions
Translate the following into numerical expressions:
- The sum of a number, eight, and seven.
- Three times the sum of a number and nine.
- Four less than four times the difference of ten and a number.
- The mean of a number, five, seven, and twelve.
- Six times the quotient of five and a number is greater than two.
- The difference of eight and a number, divided by four equals the sum of the number and two.
- The difference of two sums: eight and a number and six and another number is at least the sum of the second number and one.
- The sum of eight times a number and three times the same number is less than the difference of the number and ten.
- The difference of eleven times a number and two times another number equals twenty less than the first number.
- The total of six times a number and the same number divided by three is no more than four times the number.
Translate the following expressions, equations, and inequalities into words:
- $3x+9$
- $3(2x+1)$
- $\dfrac{4}{x}+9x$
- $\dfrac{5+x}{2}\leq 42$
- $(3x+7)-(2x+2)>6x$
- $x(5+x)-3<60$
- $x-9\geq 6x$
- $\dfrac{x}{6+x}=6x-y$
- $5x+6y>\dfrac{x}{5y}$
- $5x(\dfrac{x}{6})=7+x$