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write the following sentences as a system of inequalities in two variables ,then graph the system. The sum of the x -variable and the y- variable is at most 8. The y- variable added to the product of 3 and x-variables does not exceed 6.

User Kara Woo
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Final answer:

The two sentences as inequalities are x + y ≤ 8 and y + 3x ≤ 6. When graphing, plot the boundary lines and shade the region that satisfies both inequalities. The intersection of the shaded regions represents the solution to the system.

Step-by-step explanation:

To write the following sentences as a system of inequalities in two variables and then graph the system:

  1. The sum of the x-variable and the y-variable is at most 8. This can be represented as x + y ≤ 8.
  2. The y-variable added to the product of 3 and the x-variables does not exceed 6. This can be written as y + 3x ≤ 6.

To graph these inequalities:

  • First, plot the boundary lines for each inequality. For x + y = 8, this is a line where when x is 0, y is 8, and when y is 0, x is 8. For y + 3x = 6, when x is 0, y is 6, and when y is 0, x is 2.
  • Next, since both inequalities use '≤', you will shade the area below and to the left of each line because those are the points that satisfy each inequality.
  • The area of interest where we shade is the region where both inequalities overlap.

By creating a plot and using the given points, you will see the feasible region that satisfies both inequalities.

User ArashM
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