1. Simple inequalities Teacher notes
The Standards for Mathematical Practices describe varieties of expertise that mathematics educators at all levels should seek to develop in their students.
They are:
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for an express regularity in repeated reasoning.

2. Inequality notation
You can remember the meanings of the inequality symbols by thinking of them as a crocodile’s mouth; the crocodile will always eat the bigger number. 3 5
3 < 5
3 is less than 5 5 3
5 > 3
5 is greater than 3
Photo credit: © Kaido Kärner, Shutterstock.com 2012
Sometimes two inequalities can be combined into a single statement. For example,
If x > 3 and x ≤ 14 we can write
3 < x ≤ 14

3. Constructing inequalities

4. New York house prices Teacher notes
This activity will help students understand constraints in a real-life context by identifying the house price that meets each buyer’s requirements.
Ask students to write an inequality for each buyer’s statement (where x is an acceptable house price), e.g. 1,500,000 ≤ x ≤ 2,000,000 for the first question. Point out the use of words such as “exceed”, “at least” and “under” that indicate which inequality symbol should be used.
Mathematical Practices
4) Model with mathematics.
This activity demonstrates how students should apply the mathematics they know to solve problems arising in everyday life.
Students should use this activity to practice thinking of real-life contexts in a mathematical sense. When writing an appropriate inequality for each buyer’s constraints, students should interpret the phrasing used to decide whether it indicates the use of the symbol < or ≤.

5. Inequalities on number lines
How many values of x satisfy the inequality x > 2?
There are infinitely many values that x could take.
x could be equal to 3, , , … 3 11
It would be impossible to write every solution down.
We can visualize the solution set on a number line:
An open circle, , at 2 means that this number is not included. The arrow at the end of the line means that the solution set extends in the direction shown.

6. Inequalities on number lines
How many values of x satisfy the inequality –1 ≤ x < 4?
Justify your answer.
Although x is between two values, there are still infinitely many values that x could take. 16
x could be equal to 2, –0.7, –3 , … 17
We can represent the solution set on a number line:
Teacher notes
Students may think that the reason inequalities such as x > 2 have infinitely many solutions is because the numbers extend to infinity. It may not be obvious to students that between any two numbers there are also infinitely many real numbers. This is due to the fact that a number line is continuous with no gaps. This means that any two values, no matter how close, would always have infinitely many other values between them.
Mathematical Practices
3) Construct viable arguments and critique the reasoning of others.
This slide demonstrates how students should understand and use previously established results to construct arguments.
Having seen in the previous slides that inequalities such as x > 2 have infinitely many solutions, students should realize that the inequality on this slide will also have infinitely many solutions. They should justify this by talking about the fact that the real number line is continuous and therefore between any two numbers there are infinitely many solutions.
A solid circle, , is used at –1 because this value is included. An open circle, , is at 4 because this value is not included.
The line represents all the values in between.

7. Representing inequalities
Teacher notes
The markers on the right hand side of the interactive activity are used to amend the way the inequality is represented. Select the type of marker that is required to correctly represent the inequality. You can change the direction of the inequality once you have selected the representation you wish to use. 7

8. Inequalities in context
Sean runs a 100 m race in 20 seconds. We can draw a number line where s represents any running speed quicker than Sean’s: s
m/s
Do you think s could represent any speed greater than 5 m/s? Explain your answer.
In certain contexts, the value of x (or here, s) cannot realistically be any real number.
Teacher notes
s represents the speed, which is distance ÷ time. Here, Sean’s speed is 100 m ÷ 20 s = 5 m/s.
Mathematical Practices
2) Reason abstractly and quantitatively.
This slide demonstrates how students should make sense of quantities and their relationships.
Students should be able to contextualize; probe into the meanings of the symbols involved in the inequality. They should realize that values of s over a certain number would be unrealistic for humans and so s could not represent any speed greater than 5 m/s.
4) Model with mathematics.
This slide demonstrates how students should apply the mathematics they know to solve problems arising in everyday life.
Students should understand how the speed example translates into the number line shown on the slide. They should see that to graph s, they should first calculate distance ÷ time. They should then think about the context of the example and realize that, realistically, there are constraints on this inequality due to the limits of human running speeds.
It would be very unrealistic for a human to run any faster than about 10.5 m/s.
Usain Bolt’s world record 100 m time is 9.58 s; that’s an average speed of 10.4 m/s!

9. Integer solutions
Photo credit: © Vbar, Shutterstock.com 2012