Notes

Lesson Objectives
Now that you have completed this lesson, you should be able to say:

I can write an inequality to represent a constraint in a real-world or mathematical situation.
I can recognize the solution to an inequality has infinitely many solutions.
I can represent the solution to an inequality on a number line.
Vocabulary
Constraint: A condition or restriction in a scenario or problem that limits the possible values of the solution.

Lesson Summary
The number that is substituted for the variable and makes a mathematical sentence true is known as the solution. Inequalities have an unlimited number of solutions, which can be represented with a graph.

Solutions to inequalities are determined by the constraint, which is some condition that explains what the solutions can be to make the inequality true.

When given a scenario with a constraint, you can write inequalities and graph their solutions on a number line by following these steps:

Read the scenario to define the variable.
Rewrite the sentence to understand the constraint.
Write the inequality based on the constraint.
Graph the solution to show all possible values that make the inequality true.
Explain the solution to understand the solution as it relates to the scenario.
<
is less than
is fewer than

≤
is less than or equal to
is at most
is no more than

>
is greater than
is more than

≥
is greater than or equal to
is at least
is no less than

Inequality and Graphing Symbols:

Chart shows less than as an open circle with arrow pointing left, greater than as an open circle with arrow pointing right, less than or equal to as a closed circle with arrow pointing left, and greater than or equal to as a closed circle with arrow pointing to the right.
Examples: Pay attention to the use of an open circle versus a closed circle.

Inequality Graph
x > 3 Number line with open circle on 3 and shading to the right.
x < 3 Number line with open circle on 3 and shading to the left.
x ≥ 3 Number line with closed circle on 3 and shading to the right.
x ≤ 3 Number line with closed circle on 3 and shading to the left.
Real-World Example

The situation: The speed limit on a road is 45 miles per hour. Write and graph an inequality to represent the solution to this situation.

The variable: the speed of the car; represent it with the variable x.

Rewrite: The car's speed must be less than or equal to 45.

The inequality: x ≤ 45

To graph the inequality, draw a number line, use a closed circle on 45, and shade the line to the left.

Number line with closed circle on 45 and shading to the left.
Solution: The car's speed must be 45 miles per hour or less. However, a car cannot have a speed less than 0 miles per hour. A realistic solution set for this situation is all numbers from 0 to 45 miles per hour.