Graph the solution set for this inequality: $-6x - 3y \leq -18$
Step 1: Identify the $x$- and $y$-intercepts of the boundary line.
When $x=0, y=$ $\square$
When $y=0, x=$ $\square$
Answer (2)
Find the x and y intercepts: x = 3 , y = 6 .
Plot the intercepts and draw a solid line through them.
Test the point ( 0 , 0 ) in the inequality: 0 ≤ − 18 (False).
Shade the region that does not contain ( 0 , 0 ) .
x = 3 , y = 6 , shade the region not containing ( 0 , 0 )
Explanation
Finding Intercepts Let's start by finding the x - and y -intercepts of the boundary line − 6 x − 3 y = − 18 . These intercepts will help us graph the line.
Calculating y-intercept To find the y -intercept, we set x = 0 in the equation − 6 x − 3 y = − 18 :
− 6 ( 0 ) − 3 y = − 18 − 3 y = − 18 y = − 3 − 18 = 6 So, the y -intercept is 6.
Calculating x-intercept To find the x -intercept, we set y = 0 in the equation − 6 x − 3 y = − 18 :
− 6 x − 3 ( 0 ) = − 18 − 6 x = − 18 x = − 6 − 18 = 3 So, the x -intercept is 3.
Filling the table Now we can fill in the table:
x
y
0
6
3
0
Determining the Shaded Region Next, we need to determine the region to shade. The inequality is − 6 x − 3 y ≤ − 18 . Let's test the point ( 0 , 0 ) :
− 6 ( 0 ) − 3 ( 0 ) ≤ − 18 0 ≤ − 18 This is false, so we need to shade the region that does not contain the point ( 0 , 0 ) .
Type of Boundary Line The boundary line is solid because the inequality includes "equal to" ( ≤ ).
Summary of the Solution In summary, we have a solid boundary line passing through the points ( 3 , 0 ) and ( 0 , 6 ) . We shade the region that does not contain the origin ( 0 , 0 ) .
Examples
Understanding linear inequalities helps in various real-life situations. For instance, imagine you're managing a budget where you need to ensure your expenses ( x and y ) stay below a certain limit ( − 6 x − 3 y ≤ − 18 ). Graphing this inequality allows you to visualize the feasible spending combinations that keep you within your budget. Similarly, in manufacturing, inequalities can represent constraints on resources, helping to determine optimal production levels. This concept is also applicable in health, where inequalities can define safe ranges for medical dosages or nutritional intake, ensuring that levels are neither too high nor too low for maintaining health.
To find the solution set for the inequality − 6 x − 3 y ≤ − 18 , we identify the x -intercept as 3 and the y -intercept as 6. We plot these intercepts and draw a solid line, shading the region that does not include the origin ( 0 , 0 ) . This shaded area represents all the solutions to the inequality.
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