Choose the type of boundary line:
Solid (-)
Dashed (--)
Enter two points on the boundary line:
( [ ] , [ ] ) ( [ ] , [ ] )
Select the region you wish to be shaded:
A or B
Answer (2)
Rewrite the inequality 18"> 2 y − 6 x > 18 as 3x + 9"> y > 3 x + 9 .
Determine that the boundary line is dashed because the inequality is strict.
Find two points on the boundary line: ( 0 , 9 ) and ( − 3 , 0 ) .
Shade the region above the line because the inequality is 3x + 9"> y > 3 x + 9 .
The boundary line is dashed, two points on the line are ( 0 , 9 ) and ( − 3 , 0 ) , and the region above the line should be shaded.
Explanation
Understanding the Inequality We are given the inequality 18"> 2 y − 6 x > 18 . Our goal is to determine the characteristics of its solution set for graphing. This includes identifying the type of boundary line (solid or dashed), finding two points on the boundary line, and determining which region to shade.
Converting to Slope-Intercept Form First, let's rewrite the inequality in slope-intercept form, which is y = m x + b , where m is the slope and b is the y-intercept. We have:
18"> 2 y − 6 x > 18
Add 6 x to both sides:
6x + 18"> 2 y > 6 x + 18
Divide both sides by 2:
3x + 9"> y > 3 x + 9
Determining the Boundary Line Type Now, let's determine the type of boundary line. Since the inequality is strict ( "> > ), the boundary line will be dashed to indicate that points on the line are not included in the solution set.
Finding Two Points on the Boundary Line Next, we need to find two points on the boundary line y = 3 x + 9 . We can choose any two values for x and calculate the corresponding y values.
Let x = 0 :
y = 3 ( 0 ) + 9 = 9
So, the point ( 0 , 9 ) is on the line.
Let x = − 3 :
y = 3 ( − 3 ) + 9 = − 9 + 9 = 0
So, the point ( − 3 , 0 ) is on the line.
Determining the Shaded Region Finally, we need to determine which region to shade. Since the inequality is 3x + 9"> y > 3 x + 9 , we shade the region above the line. This is because for any given x , the y values that satisfy the inequality are greater than the y value on the line.
Final Answer In summary, the boundary line is dashed, two points on the line are ( 0 , 9 ) and ( − 3 , 0 ) , and we shade the region above the line.
Examples
Understanding linear inequalities is crucial in various real-world scenarios, such as budgeting and resource allocation. For instance, if you have a limited budget for entertainment and food, you can use linear inequalities to determine how much you can spend on each category while staying within your budget. Graphing these inequalities helps visualize the possible spending combinations that satisfy your budgetary constraints, allowing for informed decision-making.
The type of boundary line depends on whether the inequality is strict or non-strict. Two points can be calculated on the boundary line, and the shading direction will depend on the sign of the inequality. Using these principles will help in accurately graphing linear inequalities.
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