Question

The graph of inequations $x \leq y$ and $y \leq x+3$ is located in

• A. II quadrant
• B. I, II quadrants
• C. I, II, III quadrants
• D. II, III, IV quadrants

Answer 1

Answer: (C)

I, II, III quadrants

Answer 2

The given inequations are $x \leq y$ and $y \leq x+3$. This can be combined into a single inequality: $x \leq y \leq x+3$.

We need to find the region in the Cartesian plane that satisfies this inequality. The region is bounded by the lines $y = x$ and $y = x+3$.

The line $y=x$ passes through the origin (0,0) and has a slope of 1.
The line $y=x+3$ passes through (0,3) and (-3,0) and has a slope of 1. The two lines are parallel.

The inequality $y \geq x$ represents the region on or above the line $y=x$.
The inequality $y \leq x+3$ represents the region on or below the line $y=x+3$.

The combined inequality $x \leq y \leq x+3$ represents the region between the two parallel lines $y=x$ and $y=x+3$, including the lines themselves.

Now let's determine which quadrants this region occupies. The four quadrants are defined by the signs of x and y:

• Quadrant I: x > 0, y > 0
• Quadrant II: x < 0, y > 0
• Quadrant III: x < 0, y < 0
• Quadrant IV: x > 0, y < 0

Let's check if there are points in the region $x \leq y \leq x+3$ that fall into each quadrant.

• Quadrant I (x > 0, y > 0):
  Consider a point (1, 2). Here x=1, y=2.
  Check the inequalities: $1 \leq 2$ (True) and $2 \leq 1+3=4$ (True).
  Since the point (1, 2) satisfies the inequalities and is in Quadrant I, the region is located in Quadrant I.

• Quadrant II (x < 0, y > 0):
  Consider a point (-1, 1). Here x=-1, y=1.
  Check the inequalities: $-1 \leq 1$ (True) and $1 \leq -1+3=2$ (True).
  Since the point (-1, 1) satisfies the inequalities and is in Quadrant II, the region is located in Quadrant II.

• Quadrant III (x < 0, y < 0):
  Consider a point (-4, -2). Here x=-4, y=-2.
  Check the inequalities: $-4 \leq -2$ (True) and $-2 \leq -4+3=-1$ (True).
  Since the point (-4, -2) satisfies the inequalities and is in Quadrant III, the region is located in Quadrant III.

• Quadrant IV (x > 0, y < 0):
  Consider a point (1, -1). Here x=1, y=-1.
  Check the inequalities: $1 \leq -1$ (False).
  The first inequality $x \leq y$ is not satisfied for any point in Quadrant IV, because in Quadrant IV, x > 0 and y < 0, which implies y < x. The condition $x \leq y$ means x is less than or equal to y, which contradicts y < x when x > 0 and y < 0.
  Therefore, there are no points in Quadrant IV that satisfy the given inequations. The region is not located in Quadrant IV.

The graph of the inequations $x \leq y$ and $y \leq x+3$ is located in Quadrants I, II, and III.