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Learn it your way
A short class with Sarah and her students. The other ways are under More.
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Today, we're going to discuss linear inequalities! Can anyone tell me what an inequality is?
I think it’s like an equation, but instead of saying something is equal, it shows that one side is greater or less than the other.
Exactly! An inequality compares two expressions and indicates a range of possible solutions. Now, we represent these inequalities on a number line. For example, if we have 'x < 4', how do we graph it?
We put an open circle at 4 because it's not included, right?
Correct! And then we shade to the left of 4. Can anyone tell me why we shade in that direction?
Because all numbers less than 4 are included.
That's right! Remember, shading indicates all the solutions that satisfy our inequality.
So, if it was 'x ≥ 4', we would use a closed circle and shade to the right?
Exactly! Great job, everyone. Let's recap: open means 'not including,' closed means 'including.' Make sure you remember that!
Now, let’s solve an example together: how would we graph the inequality 'x - 2 ≥ 1'?
We first solve it. That gives us 'x ≥ 3.'
Great step! What does the graph look like for this inequality?
We put a closed circle on 3 and shade to the right!
Perfect! Let's try another one. Consider 'y < -1.' How do we represent this on the number line?
An open circle at -1 and shade to the left because it includes all numbers less than -1!
Exactly! Be sure to practice this at home too. Visualizing these inequalities is the key to understanding.
Let's talk about some common mistakes when graphing inequalities. What do you think is a common error students make?
Sometimes they forget to flip the circle when it's 'less than or equal to' or 'greater than or equal to.'
Good point! And how about when it comes to shading? Any tips?
I think they might accidentally shade the wrong direction!
That’s correct! Remember, you shade in the direction of the inequality. It’s essential to double-check both the circles and the shading to avoid confusion.
Can we practice with a few more examples together?
Of course! Let’s reinforce this understanding through practice.
An open circle means don't include; for shading left, be rather shrewd.
Once, a clever fox found a number line where he could only include his friends at points of closed circles and had to keep his distance from open points.
Remember 'Less = Left' helps you find where to shade!
C.O.L.E. - Circle Open means Less Excluded!
Term
What do we use for 'greater than' in graphing?
Click to flip
Term
How do we shade for 'x < 5'?
Click to flip
An inequality that compares two expressions using symbols like <, ≤, >, or ≥ rather than equality.
A graphical representation that indicates the endpoint is not included in the solution set, used for '<' or '>'.
A graphical representation showing that the endpoint is included in the solution set, used for '≤' or '≥'.
This section explains how to graph linear inequalities on the number line, highlighting the use of open and closed circles, as well as shading techniques.
In section 2.3, students learn to graph linear inequalities in one variable on the number line. The section covers the significance of using open and closed circles to represent inequality relationships and provides examples for practical understanding.
In this section, we delve into the process of graphing linear inequalities on the number line. A linear inequality indicates a range of values for a variable rather than a specific number, making it crucial to represent this concept accurately when graphing.
< or >. In contrast, a closed circle (●) is employed for inequalities that include the boundary value, such as ≤ or ≥.x < 3, you would shade all numbers to the left of 3. Conversely, if it’s x ≥ 3, you would shade to the right, including 3 itself.This method helps visually represent the set of solutions to linear inequalities, aiding students in grasping the concept that solutions can lie within a range rather than being confined to a single point.