Inequalities on Graphs worksheet

10 GCSE practice questions with answers, free to print. Every sheet is generated, so you can make a fresh one whenever you need it.

1. 2 marks

   Here is an inequality.

   \(y {sym} {m:coef}x {c:+}\)

   Does the point \(({px}, {py})\) satisfy this inequality?

   You must show your working.

2. 2 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?

3. 2 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.

4. 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.

5. 2 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 \(R\).

6. 3 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.

7. 2 marks

   Here is an inequality.

   \(y {sym} {m:coef}x {c:+}\)

   Does the point \(({px}, {py})\) satisfy this inequality?

   You must show your working.

8. 2 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 \(y \leq 3x - 4\)

   Should the boundary line \(y = 3x - 4\) be drawn solid or dashed?

   (b)

   A different region is defined by \(x \gt 3\)

   Should the boundary line \(x = 3\) be drawn solid or dashed?

9. 2 marks

   The line \(y = -3x + 3\) 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.

10. 2 marks

    A point \((x, y)\) satisfies all three of these inequalities.

    \(x \gt 2 \qquad y \lt 7 \qquad x + y \lt 11\)

    Here are four points.

    \((1, 6) \qquad (3, 6) \qquad (3, 7) \qquad (5, 6)\)

    Only one of them satisfies all three inequalities. Write down that point.

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