13. (a) On the grid show, by shading, the region that satisfies all these inequalities. $$\begin{align*} x & \geq 0 \\ x & \leq 2 \\ y & 6 \end{align*}$$ Label the region R. (b) The diagram below shows the region S that satisfies the inequalities $$\begin{align*} y & \leq 4x \\ y & > \frac{1}{2}x \\ x + y & \leq 6 \end{align*}$$ Geoffrey says that the point with coordinates (2, 4) does not satisfy all the inequalities because it does not lie in the shaded region. Is Geoffrey correct? You must give a reason for your answer.

GCSE Edexcel Maths Angles in Polygons & Parallel Lines: 13. (a) On the grid show, by sha

13. (a) On the grid show, by shading, the region that satisfies all these inequalities - Edexcel - GCSE Maths - Question 14 - 2020 - Paper 3

Question 14

13. (a) On the grid show, by shading, the region that satisfies all these inequalities. $$\begin{align*} x & \geq 0 \\ x & \leq 2 \\ y & < x + 3 \\ 2x + 3y & > 6... show full transcript

Worked Solution & Example Answer:13. (a) On the grid show, by shading, the region that satisfies all these inequalities - Edexcel - GCSE Maths - Question 14 - 2020 - Paper 3

Step 1

Part (a) - Shade the region R

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Answer

To shade the region R that satisfies the given inequalities, we proceed with the following steps:

Plotting the boundaries of each inequality:
- For the inequality $x \geq 0$ , shade to the right of the y-axis.
- For $x \leq 2$ , shade to the left of the line $x = 2$ .
- For $y < x + 3$ , draw the line $y = x + 3$ and shade below this line.
- For $2x + 3y > 6$ , rewrite it as $y > -\frac{2}{3}x + 2$ . Draw the line and shade above it.
Finding the overlapping region: The final region R is where all shaded areas overlap.
Label the region: Clearly label this overlapping region as R on the grid.

Step 2

Part (b) - Evaluate the point (2, 4)

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Answer

To determine if Geoffrey is correct regarding the point (2, 4), we need to check if this point satisfies all inequalities:

Check each inequality:
- For $y \leq 4x$ :
  Substitute (2, 4): $4 \leq 4(2) \Rightarrow 4 = 8$ (True).
- For $y > \frac{1}{2}x$ :
  Substitute (2, 4):
  $4 > \frac{1}{2}(2) \Rightarrow 4 > 1$ (True).
- For $x + y \leq 6$ :
  Substitute (2, 4):
  $2 + 4 \leq 6 \Rightarrow 6 = 6$ (True).
Conclusion:
The point (2, 4) lies on the boundary of the last inequality, $x + y = 6$ , meaning it satisfies all conditions since it includes points on the boundary for $\leq$ .

Therefore, Geoffrey is incorrect; the point (2, 4) does satisfy all inequalities.

13. (a) On the grid show, by shading, the region that satisfies all these inequalities - Edexcel - GCSE Maths - Question 14 - 2020 - Paper 3

Worked Solution & Example Answer:13. (a) On the grid show, by shading, the region that satisfies all these inequalities - Edexcel - GCSE Maths - Question 14 - 2020 - Paper 3

Part (a) - Shade the region R

Part (b) - Evaluate the point (2, 4)

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