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A short class with Sarah and her students. The other ways are under More.
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To graph a linear inequality, the first step is often to rewrite it in the form y = mx + b. Can anyone tell me what this form represents?
It shows the slope and the y-intercept!
Exactly! The slope (m) tells us how steep the line is, and the y-intercept (b) tells us where the line crosses the y-axis. Let's practice by rewriting the inequality 2x + 3y ≤ 6.
I think we need to isolate y, right?
Correct! What do we get when we do that?
We get y ≤ -1x + 2.
Nice job! Now, we can graph this inequality.
Now that we have our inequality in slope-intercept form, what's the next step?
We draw the boundary line, right? But how do we know if it's dashed or solid?
Good question! If our inequality includes ≤ or ≥, we draw a solid line, indicating that those points are included. If it uses < or >, we use a dashed line. Can anyone remind us why that is?
Because we can include the points with ≤ or ≥, but not with < or >!
Exactly! Let's go ahead and sketch the boundary line for y ≤ -1x + 2.
Now that we've drawn our boundary line, we need to test a point to decide which side to shade. Which point do you think we should test?
Let's test (0,0)!
Great choice! Plugging (0,0) into our inequality y ≤ -1(0) + 2, what do we get?
We get 0 ≤ 2, which is true!
So that means we shade the side where (0,0) is located. Who can tell me the final step after shading?
We need to ensure that our shading is clean so we can see our solution clearly!
Exactly! Always ensure your graph is neat for clear communication.
The line that defines the limit for the solution area in a graph.
Solid lines are used when the inequality includes equality (≤ or ≥), while dashed lines are for strict inequalities (< or >).
Representing the solution set on the graph by shading the appropriate area.
For less than, draw a dash, solid for more, let's make a splash!
Imagine you are a traveler. If the road is dashed, you must avoid traveling over it; if it’s solid, you are free to roam!
Remember: S for Solid lines with ≤ or ≥ and D for Dashed lines with < or >.
SAD - Solid for ≤ and ≥, Dashed for < and >.
Term
Boundary Line
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Term
Shading
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Term
Dashed Line
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This section outlines the steps necessary to graph linear inequalities in two variables.
The section details the process of graphing linear inequalities in two variables, covering how to draw boundary lines, test points, and shade the correct regions. It emphasizes the importance of understanding whether to use solid or dashed lines based on the inequality's direction.
Graphing linear inequalities involves understanding how to visually represent the range of solutions defined by the inequality. This section elaborates on the steps required for graphing inequalities in two variables, ensuring a comprehensive understanding of how to distinguish between solid and dashed lines for boundary conditions.
y = mx + c, if necessary.≥ or ≤ (indicating that points on the line are included in the solution).< or > (indicating that points on the line are not included).To graph the inequality x + y < 4:
x + y = 4, which will be drawn as a dashed line.0 + 0 = 0 < 4, indicating that we shade the region including (0,0).This method allows us to visualize the solution set of a linear inequality effectively.