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A short class with Sarah and her students. The other ways are under More.
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Today, we will explore systems of linear inequalities. Can anyone summarize what a linear inequality is?
A linear inequality compares two expressions using symbols like < or ≥.
Exactly! Now, how do you think we might deal with more than one linear inequality at a time?
We could graph both inequalities and see where they overlap.
Right again! This overlapping region is where solutions satisfy all inequalities. Let's remember it with the acronym 'S.O.L.' which stands for 'Satisfy Overlapping Lines'!
That makes sense! So the solution is the part that satisfies both inequalities.
Exactly! And we’ll look at an example of this soon.
To graph our system, we start with the inequalities, like x + y ≤ 6 and x ≥ 2. Who can tell me how we begin?
First, we need to graph the boundary line for each inequality!
Correct! Remember, a solid line is used for ≤ or ≥ and a dashed line for < or >. How would we graph x + y ≤ 6?
We would find where x and y intercept with the line and draw a solid line.
Perfect! After that, we shade the region. Who wants to explain how to determine which side to shade?
We can test a point, like (0, 0), to see if it satisfies the inequality.
Exactly! That point is very useful. Remember, if it satisfies the inequality, we shade that side.
Once we have graphed both inequalities, what do we do next?
We look for where the shaded areas overlap.
Exactly! This overlapping region is our solution. Can someone remind me how we represent this in a word problem?
We use inequalities to describe constraints, like how many items we can afford based on our budget.
Great connection! Let’s put that into practice with an example next.
You graph each inequality and find the overlapping shaded regions.
The line drawn represents the inequality's equality condition.
Parts of the graph are shaded to illustrate which values satisfy the inequalities.
The area where all valid shaded regions intersect.
To find the line, draw it right, shade the area—oh what a sight!
Imagine a market where two salesmen represent inequalities. They cannot sell beyond their limits, but finding the overlap helps them meet customer needs.
S.O.L. - Satisfy Overlapping Lines!
G.R.A.S.P. - Graph, Result, Analyze, Shade, Prove - the steps to working with inequalities.
Term
System of Inequalities
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Term
Boundary Line
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Shading in Graphs
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Overlap Region
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A set of two or more inequalities involving the same variables.
A line that represents the equality part of an inequality, helps in graphing it.
The process of marking the region of a graph that satisfies an inequality.
The area on a graph where the shaded regions of multiple inequalities meet.
This section introduces systems of linear inequalities, explaining how to graph multiple inequalities simultaneously to find common solution regions.
The section explores systems of linear inequalities, detailing how to represent these inequalities graphically. It explains the process of finding overlapping solution areas for multiple inequalities and emphasizes their application in solving real-world problems.
In this section, we discuss systems of linear inequalities, which involve two or more inequalities that are considered at the same time. The solution to such a system is the region where the shaded areas of the graphed inequalities overlap. We begin by learning how to graph each inequality individually, determine the corresponding shaded regions, and then identify the area where these regions intersect.