Minds On

Comparing

Consider the following numbers: 5 and 6.

How do they compare?

You could say that 5 is less than 6, and 6 is 1 more than 5.

Now, consider the following list of numbers and how they compare. Record your answers using a method of your choice.

6 and 12
Numbers 1, 2, 3, 4, 5, and 8
6 and 6

Explore the following number lines:

Description

A chart showing how different symbols are used to indicate inequalities and how they are represented on a number line.

The greater than symbol is 2 connected line segments forming the shape of an arrow pointing to the right. Number line example: x greater than 5 is demonstrated by an unfilled circle above the line at 5 and an arrow that points along the number line to the right.

The less than symbol is 2 connected line segments forming the shape of an arrow pointing to the left. Number line example: x less than negative 1 is demonstrated with an unfilled circle above the line at negative 1 and an arrow that points along the number line to the left.

The greater than or equal to symbol is 2 connected line segments forming the shape of an arrow pointing to the right and a horizontal line right under it. Number line example: X greater than or equal to 3 demonstrated with a filled circle above the line at 3 and an arrow that points along the number line to the right.

The less than or equal to symbol is 2 connected line segments forming the shape of an arrow pointing to the left and a horizontal line right under it. Number line example: x less than or equal to 5 is demonstrated with a filled circle above the line at 5 and an arrow that points along the number line to the left.

What are some similarities between the number lines?

Do you notice any differences?

What do you think the differences could mean?

Action

Comparing numbers

As you may have noticed, the numbers in the Minds On Section can be connected in many ways. One way that we can describe them uses terms such as equal, greater than, and less than.

For example:

a) 5 is greater than 2.

Number line from 0 to 10. Numbers 2 and 5 are in bold.

b) or $5 + 3$ is greater than 4.

$5 + 3 = 8$

Therefore 8 is greater than 4.

A 0 to 10 number line. Numbers 4 and 8 in bold. 5 plus 3 are shown as skip counts ending at 8.

We can also use symbols: $> < =$

$>$ represents greater than (e.g., $8 > 4$ which is read 8 is greater than 4).
$<$ represents less than.
$=$ represents equal.
$\geq$ represents greater than or equal to.
$\leq$ represents less than or equal to.

When we show 2 expressions or values that are not equal, for example $8 > 4$ or $2 + 10 < x + 5$

it is called an inequality.

Sometimes an inequality can represent many numbers.

$x \geq 3 + 2$

$x \geq 5$

Therefore, x is any number greater than or equal to 5, meaning 5, 6, 7, 8, 9, 10, … 21, ....

Let’s graph the solution.

Here are the number lines you examined in the Minds On Section.

The anchor chart provided indicates how inequalities can be graphed.

Notice that greater than and less than have an open dot over the number.

This is because the number is not included in the solution.

If we are considering $x > 5$ , x cannot be 5, but it could be anything that’s larger than 5 like 5.1 or even 5.01.

Description

There’s a number line representing x is greater than 5, with an open circle around 5 and an arrow that stretches to the right demonstrating that x can be anything greater than 5, but not 5.

1) Solve and graph the following on a number line.

$3 > c$
$x > 3$
$t \leq 25$

You can use the following printable Inequalities on a Number Line document to record your answers. You can also use your notebook or a voice recorder to describe the solution.

Press the ‘Activity’ button to access Inequalities on a Number Line.

Activity

Sometimes we need to solve an inequality.

Reflect back to some strategies explored to solve equations like using our number sense, building a model or number line, guess and check, or using the opposite (inverse) operation.

\begin{aligned} Solve for x + 3 & \leq 12 . \\ x + 3 - 3 & \leq 12 - 3 \\ x & \leq 12 - 3 \\ x & \leq 9 \end{aligned}

Now check some numbers to see if our value for x is true.

$5$ is $\leq 9$

Substitute 5 into the inequality for $x$ .

$5 + 3 \leq 12$

$8 \leq 12$

so yes, 5 works!

How about 9?

Substitute 9 into the inequality for $x .$

$9 + 3 \leq 12$

$12 \leq 12$

so yes, 12 works!

Now, show the inequality on a graph.

Number line 0 to 10 with line that stretches from 9 on the right all the way to the left. There's a closed circle around 9.

Solve for $2 t > 4 .$

\begin{aligned} 2 t & > 4 \\ \frac{2 t}{2} & > \frac{4}{2} \\ t & > 2 \end{aligned}

Now check some numbers to see if our value for $t$ is true.

$3 > 2$

Substitute 3 into the inequality for $t$ .

$2 (3) > 4$

$6 > 4$

so yes, 3 works!

How about 2?

Substitute 2 into the inequality for $t$ .

$2 (2) > 4$

No, 2 doesn’t work, the numbers must be greater than 2.

Now, show the inequality on a graph.

Number line 0 to 10 with line that stretches from 2 on the left all the way to the right. There's an open circle at number 2.

Your turn

Solve, verify and graph the following:

a) $x + 3 < 6$

b) $x + 5 \geq 20$

c) $x > 3 + 5 + 30$

d) $x < 50 - 12$

Click ‘Reveal’ to check your answers.

Reveal

a) $x < 3$

A number line that shows an open dot over 3, a line extended to the left over all the numbers on the number line that are lower than 3 and an arrow at the very end that is pointing to the left, away from 3, indicating that $x$ could be anything that is lower than 3 (but not 3 - hence the open dot).

b) $x \geq 15$

A number line that shows a closed dot over 15, a line extended to the right over all the numbers on the number line that are greater than 15 and an arrow at the very end that is pointing to the right, away from 15, indicating that $x$ could be anything that is bigger than 15 (including 15 - hence the closed dot).

c) $x > 38$

A number line that shows an open dot over 38, a line extended to the right over all the numbers on the number line that are bigger than 38 and an arrow at the very end that is pointing to the right, away from 38, indicating that $x$ could be anything that is bigger than 38 (not including 38 - hence the open dot).

d) $x < 38$

A number line that shows an open dot over 38, a line extended to the left over all the numbers on the number line that are smaller than 38, and an arrow at the very end that is pointing to the left, away from 38, indicating that $x$ could be anything that is lower than 38 (not including 38 - hence the open dot).

Consolidation

Solving for $x$

Complete the following questions. Record your answers using a method of your choice.

Select three questions and solve for $x .$

$x > 4$
$x + 5 > 10$
$x - 20 \geq 10$
$x + 29 \leq 32$

Click ‘reveal’ to check your answers.

Reveal

$x > 4$
a number line that shows an open dot over 4, a line extended to the right over all the numbers on the number line that are greater than 4 and an arrow at the very end that is pointing to the right, away from 4, indicating that x could be anything that is bigger than 4 (but not 4 - hence the open dot).
$x > 5$
a number line that shows an open dot over 5, a line extended to the right over all the numbers on the number line that are greater than 5 and an arrow at the very end that is pointing to the right, away from 5, indicating that x could be anything that is bigger than 5 (but not 5 - hence the open dot).
$x \geq 30$
a number line that shows a closed dot over 30, a line extended to the right over all the numbers on the number line that are greater than 30 and an arrow at the very end that is pointing to the right, away from 30, indicating that x could be anything that is bigger than 30 (including 30 - hence the closed dot).
$x \leq 3$
a number line that shows a closed dot over 3, a line extended to the left over all the numbers on the number line that are less than 3 and an arrow at the very end that is pointing to the left, away from 3, indicating that x could be anything that is less than 3 (including 3 - hence the closed dot).

Think about your learning

What is important to remember when using less than, less than or equal to, greater than and greater than or equal to symbols when comparing expressions?
What strategies do you use when solving inequalities and proving they work?
What is important to remember when graphing inequalities on a number line?

Reflection

As you read through these descriptions, which sentence best describes how you are feeling about your understanding of this learning activity? Press the button that is beside this sentence.

I feel...

I have an excellent understanding of the concepts I have a very good understanding of the concepts I have some understanding but need to review some more my understanding of the concepts needs a lot more work

Now, record your ideas using a voice recorder, speech-to-text, or writing tool.

Learning goals

Success criteria

Minds On

Comparing

Action

Comparing numbers

Your turn

Consolidation

Solving for $x$

Think about your learning

Reflection

I feel...

Learning goals

Success criteria

Minds On

Comparing

Action

Comparing numbers

Your turn

Consolidation

Solving for x

Think about your learning

Reflection

I feel...

Solving for $x$