How do you graph the inequality # 3x + 4y<12#, #x + 3y<6#, #x>=0#, #y>=0#?

Answer 1

Assumption: You do not need to be shown how to manipulate equation.

It is a matter of determining the area or areas for which all these conditions are true. This/these areas are shaded on the graph.

Note that if the inequality includes the equals then the line is solid. On the other hand if inequality does not include the equals then the line is dotted.

So #3x+4y<12 ->" dotted line "-> y<-3/4x+3#
#" "x+3y < color(white)(.)6->" solid line "->y<-1/3x+2#
#" "x>=0->" solid line"#
#" "y>=0->" solid line"#

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Answer 2

To graph the system of inequalities (3x + 4y < 12), (x + 3y < 6), (x \geq 0), and (y \geq 0):

  1. Graph the boundary lines for each inequality.
  2. Determine which side of each boundary line satisfies the inequality.
  3. Shade the regions that satisfy all the inequalities.

For the first inequality, (3x + 4y < 12), rewrite it in slope-intercept form: [4y < -3x + 12] [y < -\frac{3}{4}x + 3]

For the second inequality, (x + 3y < 6), rewrite it in slope-intercept form: [3y < -x + 6] [y < -\frac{1}{3}x + 2]

Graph the boundary lines (y = -\frac{3}{4}x + 3) and (y = -\frac{1}{3}x + 2), and indicate whether the lines are solid or dashed based on the inequality sign (< or ≤).

Next, determine which side of each boundary line satisfies the inequalities. Since both inequalities are "less than," shade the region below each line.

Lastly, since (x \geq 0) and (y \geq 0), the solution must be in the first quadrant. Shade the region that lies in the first quadrant.

The shaded region where the shaded regions from both inequalities overlap and are in the first quadrant is the solution to the system of inequalities.

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Answer 3

To graph the system of inequalities (3x + 4y < 12), (x + 3y < 6), (x \geq 0), and (y \geq 0), follow these steps:

  1. Graph the boundary lines for each inequality, excluding the inequalities with the "greater than" sign.

    • For (3x + 4y < 12), the boundary line is (3x + 4y = 12).
    • For (x + 3y < 6), the boundary line is (x + 3y = 6).
  2. Since (x \geq 0) and (y \geq 0), shade the region in the first quadrant of the coordinate plane, including the area where (x \geq 0) and (y \geq 0).

  3. Determine which side of each boundary line to shade by choosing a test point.

    • For (3x + 4y < 12), a suitable test point might be ((0, 0)).
    • For (x + 3y < 6), a suitable test point might be ((0, 0)).
  4. Shade the region where the test point satisfies the inequality.

  5. The shaded region where both inequalities are satisfied represents the solution to the system of inequalities.

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Answer 4

To graph the system of inequalities (3x + 4y < 12), (x + 3y < 6), (x \geq 0), and (y \geq 0), follow these steps:

  1. Graph the boundary lines for each inequality by replacing the inequality sign with an equal sign:

    • For (3x + 4y = 12), rearrange the equation to solve for (y): (y = \frac{-3}{4}x + 3).
    • For (x + 3y = 6), rearrange the equation to solve for (y): (y = \frac{-1}{3}x + 2).
    • Since (x \geq 0) and (y \geq 0), the solutions lie in the first quadrant.
  2. Plot the boundary lines on the coordinate plane.

  3. Determine which side of each boundary line represents the solution region for the corresponding inequality.

    • For (3x + 4y < 12), shade the region below the line.
    • For (x + 3y < 6), shade the region below the line.
  4. The intersection of the shaded regions of both inequalities represents the solution to the system of inequalities.

  5. Since (x \geq 0) and (y \geq 0), the solution region is restricted to the first quadrant.

  6. The resulting shaded region represents the solution to the system of inequalities.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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