Example - 3.2
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Interactive Audio Lesson
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Defining Linear Inequalities
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Today, we're going to discuss linear inequalities. Can anyone tell me how they differ from equations?
I think inequalities show a range of values rather than a single value.
Exactly! Inequalities use symbols like < or > instead of =. For example, instead of saying x = 3, we can say x < 5. This means x can be any value less than 5. To remember this, think of the 'inequality' as a 'range of possibilities.'
What about the other symbols like ≤ or ≥?
Great question! ≤ means 'less than or equal to', while ≥ means 'greater than or equal to'. So if we say x ≤ 3, x could be 3 or any number less than 3.
Let's summarize: Linear inequalities express ranges rather than exact values. Remember: < means less than, > means greater than, ≤ means less than or equal to, and ≥ means greater than or equal to.
Solving Linear Inequalities in One Variable
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Now let's solve some linear inequalities in one variable. For example, how would we solve 2x - 3 < 5?
We can first add 3 to both sides.
Correct! What do we get after that?
2x < 8.
Yes! Now, how do we isolate x?
Divide both sides by 2 to get x < 4.
Excellent! Always remember, when you multiply or divide by a negative number, you must reverse the inequality. Let's summarize the key point: add or subtract first, then divide or multiply, and watch out for negatives!
Graphing Linear Inequalities
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Graphing is a powerful way to illustrate solutions of inequalities. If we have the inequality x < 4, how should we graph this?
We would put an open circle at 4 because it does not include 4.
Exactly! And which direction do we shade?
We shade to the left, because we're looking for any number less than 4.
Perfect! Now this visual representation is important, and you can use an open circle for < and > and a closed circle for ≤ and ≥. Let's recap: open circle for exclusive points and closed circle for included points. What did we learn?
How to graph inequalities using number lines!
Solving Systems of Linear Inequalities
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Next, let's discuss systems of linear inequalities. If we have two inequalities, like x + y ≤ 6 and x ≥ 2, where do we graph this?
We graph each inequality separately and look for overlapping regions.
That's right! First, graph the boundary line from x + y = 6, using a solid line. Then for x ≥ 2, draw a solid vertical line at x = 2.
And we shade the region where both conditions are met.
Exactly! This overlapping area represents the solution to the system. So, we shade where both inequalities' solutions intersect. Key takeaway: always look for that overlap!
Introduction & Overview
Read summaries of the section's main ideas at different levels of detail.
Quick Overview
Standard
In this section, students learn about linear inequalities, including their definitions, types, and methods of solving in one and two variables. The importance of correctly representing solutions graphically is emphasized, alongside practical applications.
Detailed
Detailed Summary
Linear inequalities are algebraic expressions that use inequality symbols (<, ≤, >, ≥) instead of the equality sign. This section provides foundational knowledge for students to recognize and work with linear inequalities in both one and two variables. It dives into how to solve these inequalities, with rules for manipulation—especially the crucial rule of reversing the inequality sign when multiplying or dividing by a negative number. In graphical representations, students learn to depict solutions on number lines and coordinate planes effectively. Additionally, the concept of systems of inequalities and their graphical interpretations is introduced, allowing students to visualize overlapping solution regions. Real-world applications are also covered, with word problems illustrating how to translate everyday situations into mathematical inequalities.
Key Concepts
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Linear Inequalities: Expressions that involve comparison rather than equality.
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Boundary Line: The line used in graphing inequalities.
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Open and Closed Circles: Symbols indicating whether endpoints are included in the solution.
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System of Inequalities: Multiple inequalities with combined solution sets.
Examples & Applications
Example: Solve 2x - 3 < 5. Solution: Adding 3 gives 2x < 8, which leads to x < 4.
Example: Graphing the inequality x + y ≤ 6 involves drawing a solid line for the boundary and shading below it.
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
When you see a line that's dash, the solution’s area you must cash!
Stories
Imagine a budget for snacks; your inequality shows how many you can have to stay within allocated funds.
Memory Tools
For signs: Less than opens to the left, like a mouth that wants to bite!
Acronyms
I GOT IT
Inequalities Geometrically Oriented Towards Intersection
representing regions.
Flash Cards
Glossary
- Linear Inequality
An inequality that involves a linear function represented with <, ≤, >, or ≥ symbols.
- Boundary Line
The line representing the equality part of a linear inequality on a graph.
- Open Circle
A graphing symbol used to indicate an endpoint that is not included in the solution set.
- Closed Circle
A graphing symbol used to indicate an endpoint that is included in the solution set.
- System of Inequalities
A set of two or more inequalities with the same variables.
Reference links
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