What if you were given a word problem like "It took the Eagle Scouts one hour to wash 3 cars. How long did it take them to wash one car?" or "The distance from the East Coast to the West Coast is more than 2500 miles."? How could you write these sentences in algebraic form? After completing this Concept, you'll be able to write equations and inequalities for situations like these.

### Watch This

CK-12 Foundation: 0107S Write Equations and Inequalities

### Guidance

In algebra, an **equation** is a mathematical expression that contains an equals sign. It tells us that two expressions represent the same number. For example, \begin{align*}y = 12x\end{align*} is an equation. An **inequality** is a mathematical expression that contains inequality signs. For example, \begin{align*}y \le 12x\end{align*} is an inequality. Inequalities are used to tell us that an expression is either larger or smaller than another expression. Equations and inequalities can contain both **variables** and **constants**.

Variables are usually given a letter and they are used to represent unknown values. These quantities can change because they depend on other numbers in the problem.

Constants are quantities that remain unchanged. Ordinary numbers like \begin{align*}2, \ -3, \ \frac{3}{4},\end{align*} and \begin{align*}\pi\end{align*} are constants.

Equations and inequalities are used as a shorthand notation for situations that involve numerical data. They are very useful because most problems require several steps to arrive at a solution, and it becomes tedious to repeatedly write out the situation in words.

Here are some examples of equations:

\begin{align*}3x - 2 = 5 \qquad x + 9 = 2x + 5 \qquad \frac{x}{3} = 15 \qquad x^2 + 1 = 10\end{align*}

To write an inequality, we use the following symbols:

> **greater than**

\begin{align*}\ge\end{align*} **greater than or equal to**

< **less than**

\begin{align*}\le\end{align*} **less than or equal to**

\begin{align*}\neq\end{align*} **not equal to**

Here are some examples of inequalities:

\begin{align*}3x < 5 \qquad 4 - x \le 2x \qquad x^2 + 2x - 1 > 0 \qquad \frac{3x}{4} \ge \frac{x}{2} - 3\end{align*}

The most important skill in algebra is the ability to translate a word problem into the correct equation or inequality so you can find the solution easily. The first two steps are **defining the variables** and **translating** the word problem into a mathematical equation.

**Defining the variables** means that we assign letters to any unknown quantities in the problem.

**Translating** means that we change the word expression into a mathematical expression containing variables and mathematical operations with an equal sign or an inequality sign.

#### Example A

*Define the variables and translate the following expressions into equations.*

a) A number plus 12 is 20.

b) 9 less than twice a number is 33.

c) $20 was one quarter of the money spent on the pizza.

**Solution**

a) **Define**

Let \begin{align*}n=\end{align*} the number we are seeking.

**Translate**

A number plus 12 is 20.

\begin{align*} n + 12 = 20\end{align*}

b) **Define**

Let \begin{align*}n=\end{align*} the number we are seeking.

**Translate**

9 less than twice a number is 33.

This means that twice the number, minus 9, is 33.

\begin{align*}2n - 9 = 33\end{align*}

c) **Define**

Let \begin{align*}m = \end{align*} the money spent on the pizza.

**Translate**

$20 was one quarter of the money spent on the pizza.

\begin{align*}20 = \frac{1}{4} m\end{align*}

Often word problems need to be reworded before you can write an equation.

#### Example B

*Find the solution to the following problems.*

a) Shyam worked for two hours and packed 24 boxes. How much time did he spend on packing one box?

b) After a 20% discount, a book costs $12. How much was the book before the discount?

**Solution**

a) **Define**

Let \begin{align*}t = \end{align*} time it takes to pack one box.

**Translate**

Shyam worked for two hours and packed 24 boxes. This means that two hours is 24 times the time it takes to pack one box.

\begin{align*}2 = 24t\end{align*}

**Solve**

\begin{align*}t = \frac{2}{24} &= \frac{1}{12} \ \text{hours}\\ \frac{1}{12} \times 60 \ \text{minutes} &= 5 \ \text{minutes} \end{align*}

**Answer**

Shyam takes 5 minutes to pack a box.

b) **Define**

Let \begin{align*}p = \end{align*} the price of the book before the discount.

**Translate**

After a 20% discount, the book costs $12. This means that the price minus 20% of the price is $12.

\begin{align*}p - 0.20p = 12\end{align*}

**Solve**

\begin{align*}p - 0.20p &= 0.8p, \ \text{so} \ 0.8p = 12\\ p &= \frac{12}{0.8} = 15\end{align*}

**Answer**

The price of the book before the discount was $15.

**Check**

If the original price was $15, then the book was discounted by 20% of $15, or $3. \begin{align*}\$15 - 3 = \$12\end{align*}. **The answer checks out**.

#### Example C

Define the variables and translate the following expressions into inequalities.

a) The sum of 5 and a number is less than or equal to 2.

b) The distance from San Diego to Los Angeles is less than 150 miles.

c) Diego needs to earn more than an 82 on his test to receive a \begin{align*}B\end{align*} in his algebra class.

d) A child needs to be 42 inches or more to go on the roller coaster.

**Solution**

a) **Define**

Let \begin{align*}n =\end{align*} the unknown number.

**Translate**

\begin{align*}5 + n \le 2\end{align*}

b) **Define**

Let \begin{align*}d =\end{align*} the distance from San Diego to Los Angeles in miles.

**Translate**

\begin{align*}d < 150\end{align*}

c) **Define**

Let \begin{align*}x = \end{align*} Diego’s test grade.

**Translate**

\begin{align*}x > 82\end{align*}

d) **Define**

Let \begin{align*}h = \end{align*} the height of child in inches.

**Translate:**

\begin{align*}h \ge 42\end{align*}

Watch this video for help with the Examples above.

CK-12 Foundation: Write Equations and Inequalities

### Vocabulary

- To write an inequality, we use the following symbols:

> **greater than**

\begin{align*}\ge\end{align*} **greater than or equal to**

< **less than**

\begin{align*}\le\end{align*} **less than or equal to**

\begin{align*}\neq\end{align*} **not equal to**

### Guided Practice

*Define the variables and translate the following expressions into inequalities.*

a) Jose took 3 train trips in a day, some of which cost $2.75 and some of which cost $3.95. His total cost was $9.45.

b) The product of 3 and some number is more than the sum of 24 and that number.

**Solution:**

a) Let \begin{align*}t\end{align*} be the number of train rides that cost $2.75. Then \begin{align*}5-t\end{align*} is the number of train rides that cost $3.95. Then we get:

\begin{align*}2.75t + 3.95(5-t)=9.45.\end{align*}

b) Let \begin{align*}n\end{align*} be "some number." Then the product of 3 and \begin{align*}n\end{align*} is \begin{align*}3n\end{align*}. The sum of 24 and \begin{align*}n\end{align*} is \begin{align*}24+n\end{align*}. Together we get:

\begin{align*} 3n > 24+n\end{align*}

### Practice

For 1-10, define the variables and translate the following expressions into equations.

- Peter’s Lawn Mowing Service charges $10 per job and $0.20 per square yard. Peter earns $25 for a job.
- Renting the ice-skating rink for a birthday party costs $200 plus $4 per person. The rental costs $324 in total.
- Renting a car costs $55 per day plus $0.45 per mile. The cost of the rental is $100.
- Nadia gave Peter 4 more blocks than he already had. He already had 7 blocks.
- A bus can seat 65 passengers or fewer.
- The sum of two consecutive integers is less than 54.
- The product of a number and 3 is greater than 30.
- An amount of money is invested at 5% annual interest. The interest earned at the end of the year is greater than or equal to $250.
- You buy hamburgers at a fast food restaurant. A hamburger costs $0.49. You have at most $3 to spend. Write an inequality for the number of hamburgers you can buy.
- Mariel needs at least 7 extra credit points to improve her grade in English class. Additional book reports are worth 2 extra credit points each. Write an inequality for the number of book reports Mariel needs to do.