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Words that Describe Patterns

Use variable expressions to solve real-world problems.

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Words that Describe Patterns

Suppose 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 describe these situations using mathematics in order to solve them using Algebra? 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. An equation indicates that two expressions represent the same value, like y=12x,where y and 12x are equal to each other. An inequality is a mathematical expression that contains inequality signs. Inequalities are used to tell us that an expression is either larger or smaller than another expression. For example, y<12x indicates that the value of y is less than the value of 12x. Equations and inequalities can contain both variables and constants.

Variables are usually identified by a letter and 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. Numbers like 2,3,34, and π are examples of constants, because they always represent the same value.

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:

3x2=5x+9=2x+5x3=15x2+1=10

To write an inequality, we use the following symbols:

> greater than

greater than or equal to

< less than

less than or equal to

not equal to

Here are some examples of inequalities:

3x<54x2xx2+2x1>03x4x23

Arguably the most important skill in algebra is the ability to translate a word problem into an appropriate 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 problem into a mathematical statement containing variables and mathematical operations, generally with an equal sign or an inequality sign.

Example A

Define the variables and translate the following statements into equations. Notice that "is" and "was" indicate an equals or inequality sign.

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 n= the number we are seeking.

Translate

A number plus 12 is 20.

n+12=20

b) Define

Let n= the number we are seeking.

Translate

9 less than twice a number is 33.

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

2n9=33

c) Define

Let m= the money spent on the pizza.

Translate

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

20=14m

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 t= 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.

2=24t

Solve

t=224112×60 minutes=112 hours=5 minutes

Answer

Shyam takes 5 minutes to pack a box.

b) Define

Let p= 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.

p0.20p=12

Solve

p0.20p0.8pp=120.8=12=12=15

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.

$153=$12.The answer checks out.

Example C

Define the variables and translate the following statements 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 82 points on his test to receive a B in his algebra class.

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

Solution

a) Define

Let n= the unknown number.

Translate

5+n2

b) Define

Let d= the distance from San Diego to Los Angeles in miles.

Translate

d<150

c) Define

Let x= Diego’s test grade.

Translate

x>82

d) Define

Let h= the height of child in inches.

Translate:

h42

Watch this video for help with the Examples above.

CK-12 Foundation: Write Equations and Inequalities

Guided Practice

Define the variables and translate the following statements into inequalities.

a) Jose took 5 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 t be the number of train rides that cost $2.75. Then 5t is the number of train rides that cost $3.95. Then we get:

2.75t+3.95(5t)=9.45.

b) Let n be "some number." Then the product of 3 and n may be written as 3n. The sum of 24 and n may be written as 24+n.Since the statement specifies that the product is greater than the sum, we get:

3n>24+n

Explore More

For questions 1-10, define the variables and translate the following statements into equations or inequalities.

  1. Peter’s Lawn Mowing Service charges $10 per job plus $0.20 per square yard. Peter earns $25 for a job.
  2. Renting the ice-skating rink for a birthday party costs $200 plus $4 per person. The rental costs $324 in total.
  3. Renting a car costs $55 per day plus $0.45 per mile. The cost of the rental is $100.
  4. Nadia gave Peter 4 more blocks than he already had. He already had 7 blocks.
  5. A bus can seat 65 passengers or fewer.
  6. The sum of two consecutive integers is less than 54.
  7. The product of a number and 3 is greater than 30.
  8. 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.
  9. 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.
  10. 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 representing the number of book reports Mariel needs to do.

Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 1.7. 

Vocabulary

\ge

\ge

The greater-than-or-equal-to symbol "\ge" indicates that the value on the left side of the symbol is greater than or equal to the value on the right.
\le

\le

The less-than-or-equal-to symbol "\le" indicates that the value on the left side of the symbol is lesser than or equal to the value on the right.
\ne

\ne

The not-equal-to symbol "\ne" indicates that the value on the left side of the symbol is not equal to the value on the right.
constant

constant

A constant is a value that does not change. In Algebra, this is a number such as 3, 12, 342, etc., as opposed to a variable such as x, y or a.
Equation

Equation

An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs.
greater than

greater than

The greater than symbol, >, indicates that the value on the left side of the symbol is greater than the value on the right.
greater than or equal to

greater than or equal to

The greater than or equal to symbol, \ge, indicates that the value on the left side of the symbol is greater than or equal to the value on the right.
inequality

inequality

An inequality is a mathematical statement that relates expressions that are not necessarily equal by using an inequality symbol. The inequality symbols are <, >, \le, \ge and \ne.
less than

less than

The less-than symbol "<" indicates that the value on the left side of the symbol is lesser than the value on the right.
less than or equal to

less than or equal to

The less-than-or-equal-to symbol "\le" indicates that the value on the left side of the symbol is lesser than or equal to the value on the right.
not equal to

not equal to

The "not equal to" symbol, \ne, indicates that the value on the left side of the symbol is not equal to the value on the right.
Variable

Variable

A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n.

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