Minds On

Representing inequalities

Greater than, less than, equal to

A greater than symbol, a less than symbol, and an equal to symbol

Answer the following questions. You can record your ideas digitally, orally, in print, or in an organizational tool of your choice:

  • Where have you been aware of the terms “greater than,” “less than” or “equal to” in your life? <   >   =
  • What do the following symbols mean: <   >   =

Types of inequalities

Consider the following example:

Student A and Student B are collecting sea glass on the beach. At the end of the afternoon, they compared notes as to how much they each collected.

Student A had more sea glass than Student B. We can represent this as an inequality:

A > B (A is greater than B)

This is called an inequality because the amounts are not equal.

The two most common inequalities are greater than and less than, as displayed in the following chart:

Symbol

Words

Example

>

greater than

5 > 2

<

less than

7 < 9

We can also have inequalities that include “equals.” These are displayed in the following chart:

Symbol

Words

Example

greater than or equal to

x 1

less than or equal to

y 3

Respond to the following questions digitally, orally or in print:

How can you use these symbols to express a mathematical statement?

How would you express that one number is “greater than,” “less than,” or “equal to” in numbers?

What are some possibilities for the value of x in the following mathematical inequalities?

  • x > 9 (x is greater than 9)
  • x < -6 (x is less than -6)

Action

Equations and inequalities

What is an equation?

An equation is a statement that the values of two mathematical expressions are equal (indicated by the “=” sign).

An example of an equation is:

2 x   +   5   =   13

That means that 2 multiplied by a number, added with the 5 will equal 13.

For example, let’s substitute x   for   4 :

2 ( 4 )   +   5   =   13 8   +   5   =   13

What is an inequality?

An inequality, as we saw in the Minds On section, is when an expression on the one side DOES NOT equal what is on the other side of the symbol.

An example of an inequality is:

2 x   +   5   >   13

What does that mean?

That means that 2 multiplied by a number added with 5 is greater than 13.

For example, let’s substitute 5 for   x   and determine if this is true:

2 ( 5 )   +   5 >   13     10   +   5 >   13              15 >   13

Like terms and simplest form

Consider the following inequality expression:

2 x   +   3 x   >   13

In this expression, we have 2 x   ’s AND we have 3 more x   ’s.

Can we put them together?

These are called like terms.

Because they are both different amounts of x   ’s, we can put them together to make 5 x   ’s.

We can only simplify terms if they have the same variable.

Let’s practice. In the following equation there are like terms. Simplify this equation by collecting the like terms.

6 y   +   2 x     3 y   < 20

Record your response using a method of your choice.

When you are ready, press the ‘Answer’ button to reveal the suggested answer. Compare your response to the suggested answer.

If you said   6 y     3 y    can be combined as like terms you would be correct.

6 y   and   3 y    have the same variable, they can be simplified. Take note that there is a “−” sign before the   3 y ,    so that means - 3 y .

The simplified inequality would be: 3 y   + 2 x   < 20

Your turn

Reduce the following two expressions to their simplest forms. Be sure to record your steps! Record your response digitally, orally or in print.

  • 10 x     8 x     2
  • 3 x   +   10 x     4 x

Solving inequalities with one variable

We are going to begin to solve some inequalities with one variable. To do this, we work very similarly to how we solve equations.

As an example, let’s explore the following inequality:

3 x   +   3   >   9

Our goal is to solve for x   and find the value of x .
We need to isolate x   by moving the 3 (the whole number) to the other side of the equation.

3 x   +   3     3   >   9     3

3 x 3 > 6 2

x > 2 ,  

which is saying that in this case, the possibilities for x   are any number greater than 2.

Your turn

Choose two of the following inequalities to solve. Record your ideas digitally, orally, in print, or in an organizational tool of your choice.

  • x 3 < 12
  • 6 x - 3 > 9
  • 2 x + 6 < 24
  • 8 x > 32

Negative variable solutions

When we solve for a variable (an unknown number) we have to ensure that our solution has a positive variable.

So, what happens if we have a negative variable?

Consider the following expression:

7     x   <   1

Let’s isolate x   first:

7     7     x   <   1

- x   <   1     7

- x  

is the same as - 1 x .

Now we have - 1 x < - 8  
which means that we have a common factor of - 1 .

- 1 x - 1 < - 8 - 1

A rule in this case is that whenever you divide both sides of an inequality by a negative number, the inequality sign is flipped to make the statement true.

Therefore, in this case, our solution will now read x > 8 .

Your turn

Now, solve for x   in two of the following expressions.

Record your ideas digitally, orally, in print, or in an organizational tool of your choice.

Remember to show your work.

  • 9     x   <   3
  • x     6   >   4
  • 6 - 3 x < - 2
  • - x 4 > - 16

Brainstorm

Brainstorm

What do you notice in your solutions to these expressions?

Simplifying inequalities with one variable

In the previous section, we learned about how we can reduce an inequality to its simplest terms by combining like terms (terms with the same variable).

If the like terms are not on the same side of the inequality sign, we must rearrange the terms by the same method we use with other integers and variables.

7 x   +   12   >   9 x

7 x     7 x   +   12   >   9 x     7 x

12   >   2 x

12 2 > 2 x 2

6 > x

Let’s determine if this solution is correct:

7 x   +   12   >   9 x

Let’s choose a number to determine if this is correct. The solution we have determined is that 6 is bigger than x ,   or this could be stated that x   is less than 6

so let’s choose to test the value x = 2

7 ( 2 )   +   12   >   9   ( 2 )

14   +   12   >   18

26   >   18

This proves that the solution is correct.

Consolidation

Practise what you have learned

You are going to practise what you have learned in this learning activity.

Turn the three word problems into mathematical inequality sentences. Record your responses using a method of your choice.

  • Ten students collected sea glass at the beach. We don’t know how much each student collected, but we do now that all together they collected more than 60 pieces of sea glass.
  • The longest side of an irregular quadrilateral is twice the shortest side. A third side is 3 cm longer than the shortest side. The fourth side is 1 2   the length of the longest side. The perimeter of the shape is greater than 163 cm.
  • A farmer is planting fruit trees. They have space to plant no more than 43 trees. They want to plant 3 times as many avocado trees as guava trees. They have decided to also plant lemon trees. There will be 2 times as many lemon trees as there are avocado and guava combined.

Bringing it all together

We have learned new concepts today. We have learned about:

  • inequalities
  • like terms and simplifying
  • negative variables

For each of the new concepts that you have learned, let’s reflect on strengths, needs and next steps.

Respond to the following tasks. You can record your thinking digitally, orally, or in print.

  • Rank the 3 new concepts (inequalities, like terms/simplifying, and negative variables) in order of your comfort with discussing and continuing to practice. Your top concept will be first.
  • For the new concepts that you ranked at the bottom of this list, what can you do to strengthen these rankings? How can you continue to grow in your confidence for these concepts?
  • For the concepts that you ranked 1 and 2, try the following:
  • Create a practice question for someone to try (be sure to have the response ready)
  • Describe how you would help them ensure that they arrive at the correct answer

Reflection

As you read the following descriptions, select the one that best describes your current understanding of the learning in this activity. Press the corresponding button once you have made your choice.

I feel...

Now, expand on your ideas by recording your thoughts using a voice recorder, speech-to-text, or writing tool.

When you review your notes on this learning activity later, reflect on whether you would select a different description based on your further review of the material in this learning activity.