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1.1.3 Understanding Constraints

Algebra 11.1.3 Understanding Constraints

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Activity

A constraint is something that limits what is possible or reasonable in a situation.

For example, one constraint in a pizza party might be the number of slices of pizza each person could have, $s$ . We can write $s < 4$ to say that each person gets fewer than 4 slices.

Look at the expressions you wrote when planning the pizza party earlier. Choose an expression that uses one or more letters.

1.

What is the expression you are examining?

Your answer may vary, but here is an example.

$(1.50 t + 21.99) p + d$

2.

What does the first letter represent?

Your answer may vary, but here is an example.

$t$ = number of toppings

3.

What values would be reasonable for this first letter? (For instance, could the value be greater than 50? Is it possible for the letter to be a non-whole number? A negative number?)

Your answer may vary, but here is an example.

The cost of a topping is usually less than $2.00 per topping

4.

What does the second letter represent?

Your answer may vary, but here is an example.

$p$ = number of pizzas ordered

5.

What values would be reasonable for this second letter? (For instance, could the value be greater than 50? Is it possible for the letter to be a non-whole number? A negative number?)

Your answer may vary, but here is an example.

The number of pizzas ordered will probably be less than or equal to the number of students in the class.

6.

Write equations or inequalities that represent some constraints in your pizza party plan. If a quantity must be an exact value, use the = symbol. If it must be greater than or less than a certain value to be reasonable, use the $>$ or $<$ symbol.

Your answer may vary, but here is an example.

$t \leq 2.00$

$p \leq 30$

$d \leq 5.00$ if $d$ represents the delivery charge

Video: Writing an Inequality for the Constraint

Watch the following video to learn more about constraints.

Access multimedia content

Self Check

Marcus works

s

hours per week stocking shelves. He works

m

hours per week mowing lawns. Marcus works at least 8 hours per week stocking shelves. He works at most a total of 20 hours per week. Which of the following represent constraints for this situation?

$s + m \le 20; s \le 8$
$s + m \ge 20; s \le 8$
$s + m \le 20; s \ge 8$
$s + m \ge 20; s \ge 8$

Additional Resources

Inequalities

One way to represent a constraint is to use an inequality symbol. For example, you may decide that at most 3 toppings should be on each pizza. This could be represented by t < 3, if $t$ represents the number of toppings.

On the number line, the numbers get larger as they go from left to right. The number line can be used to explain the symbols “ $<$ ” and “ $>$ .”

$a < b$ is read “ $a$ is less than $b$ .” $a$ is to the left of $b$ on the number line.

$a > b$ is read “ $a$ is greater than $b$ .” $a$ is to the right of $b$ on the number line.

The expressions $a < b$ or $a > b$ can be read from left to right or right to left, though in English we usually read from left to right.

For example, $7 < 11$ is equivalent to $11 > 7$ .

For example, $17 > 4$ is equivalent to $4 < 17$ .

Inequality symbols	Words
$a \neq b$	$a$ is not equal to $b$
$a < b$	$a$ is less than $b$
$a \leq b$	$a$ is less than or equal to $b$
$a > b$	$a$ is greater than $b$
$a \geq b$	$a$ is greater than or equal to $b$

Let’s look at some specific examples:

1.

$17 \leq 26$

Compare your answer:

17 is less than or equal to 26

2.

$12 > 27 \div 3$

Compare your answer:

12 is greater than 27 divided by 3

3.

$y + 7 < 19$

Compare your answer:

y plus 7 is less than 19

Try it

Try It: Inequalities

Translate the following statements into English phrases.

1.

$14 \leq 27$

Compare your answer:

14 is less than or equal to 27

2.

$12 > 4 \div 2$

Compare your answer:

12 is greater than 4 divided by 2

3.

$x - 7 < 1$

Compare your answer:

x minus 7 is less than 1

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- Authors: Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis, Jay Abramson, Sharon North, Amy Baldwin, Alyssa Howell
- Publisher/website: OpenStax
- Book title: Algebra 1
- Publication date: Jun 4, 2025
- Location: Houston, Texas
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- Section URL: https://openstax.org/books/algebra-1/pages/1-1-3-understanding-constraints

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