Inequalities on Graphs
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Either paper 2 marks
Here is an inequality.
\(y \geq -3x + 2\)
Does the point \((-5, 19)\) satisfy this inequality?
You must show your working.
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Either paper Higher2 marks
A region on a coordinate grid is drawn by ruling its boundary lines first. A boundary line is drawn solid when the points on the line itself belong to the region, and dashed when they do not.
(a)A region is defined by \(y \gt 3x - 8\)
Should the boundary line \(y = 3x - 8\) be drawn solid or dashed?
(b)A different region is defined by \(x \lt 2\)
Should the boundary line \(x = 2\) be drawn solid or dashed?
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Either paper Higher2 marks
The line \(y = 2x + 5\) is drawn on the grid.
(a)State the inequality that is satisfied by every point below the line, but not by any point on the line itself.
(b)State the inequality that is satisfied by every point above the line and by every point on the line.
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Either paper 2 marks
A point \((x, y)\) satisfies all three of these inequalities.
\(x \gt 5 \qquad y \lt 4 \qquad x + y \lt 11\)
Here are four points.
\((6, 1) \qquad (4, 1) \qquad (6, 4) \qquad (8, 3)\)
Exactly one of them satisfies all three inequalities.
Write down that point.
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Either paper Higher2 marks
The vertical line \(x = -2\) and the line \(y = -x - 1\) are both drawn on the grid.
Region \(R\) is made up of every point that lies to the left of the line \(x = -2\) and above the line \(y = -x - 1\). No point of \(R\) lies on either line.
Write down the two inequalities that define \(R\).
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Either paper Higher3 marks
\(x\) and \(y\) are integers.
\(x \geq 1 \qquad y \geq -1 \qquad x + y \leq 3\)
(a)Work out how many points \((x, y)\) satisfy all three inequalities.
(b)Write down the point that has the greatest \(x\)-coordinate.
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Either paper 4 marks
The three unlabelled points on the grid are the corners of a triangle, and the points \(P\) and \(Q\) lie on the sides of this triangle. The region \(S\) is defined by these three inequalities.
(1) \(x \gt 0\)
(2) \(y \gt -2\)
(3) \(2x + y \leq 4\)
(a)Mina says that the point \(P\) with coordinates \((0, 3)\) does satisfy all the inequalities.
Is Mina correct? You must give a reason for your answer.
(b)Geoffrey says that the point \(Q\) with coordinates \((2, -2)\) does not satisfy all the inequalities.
Is Geoffrey correct? You must give a reason for your answer.
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Either paper Higher2 marks
On the grid, show the region that satisfies \(y ≥ -2x + 1\)
Draw the boundary line, then mark a point inside the region with the letter R.
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Either paper Higher2 marks
Joe draws this graph to identify the region \(R\) represented by
\(y ≤ x - 2\)
\(y ≤ -2 - x\)
\(x < 3\)
Joe has shaded the region \(R\).
Make two criticisms of his graph.
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Either paper Higher4 marks
The region \(R\) is shown on the grid. It is bounded by four straight lines, and the points on these lines are part of \(R\).
Find the four inequalities that define the region \(R\).
- Yes
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(a) Dashed
(b) Dashed -
(a) \(y \lt 2x + 5\)
(b) \(y \geq 2x + 5\) - \((6, 1)\)
- \(x \lt -2\) and \(y \gt -x - 1\)
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(a) \(10\)
(b) \((4, -1)\) -
(a) No. At \((0, 3)\) inequality (1) has left-hand side \(0\), and it does not allow the two sides to be equal, so the point fails it and does not satisfy all three inequalities.
(b) Yes. At \((2, -2)\) inequality (2) has left-hand side \(-2\), and it does not allow the two sides to be equal, so the point fails it and does not satisfy all three inequalities. - The line \(y = -2x + 1\) drawn, and the region above it marked
- (1) The line for the first inequality is drawn dashed, but it should be solid because the inequality includes equals. (2) The shading is on the wrong side of the line for the second inequality: the region should be below the line.
- \(x \geq -4\), \(y \geq -2\), \(y \leq 4\) and \(y \geq 3x + 1\)
Takes a couple of seconds the first time a sheet is made.