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Inequalities on graphs GCSE lesson: regions, dashed or solid lines, integer points

Name: ____________________ Date: ____________________

Show your working. Check every root by substitution and label your sketches.

Inequalities on graphs: practice

Show your working on paper: test a point for each side, and say whether each line is dashed or solid. The stretch is optional.

Diagnostic question: choose and explain

  1. How many points with whole-number coordinates are in the region x > 1, y ≥ 1 and x + y < 5?
    1. 3
    2. 6
    3. 1
    4. 10

Practice questions

  1. Is the point (2, 3) in the region y < 2x + 1?
    1. Yes: 3 < 5 is true
    2. No: 3 < 5 is false
    3. No: it is on the line
  2. The boundary line of the region x + 2y ≥ 4 is drawn…
    1. solid
    2. dashed
  3. Write the inequality for the region to the left of the dashed line x = 3.
  4. Which side of the line 3x + 2y = 12 is in the region 3x + 2y < 12?
    1. The side with (0, 0)
    2. The side without (0, 0)
    3. Neither: only the line itself
  5. The region −y < 2x − 3 is on which side of the line y = 3 − 2x?
    1. Above it
    2. Below it
  6. To shade the region y > 3x, can you test the point (0, 0)?
    1. No: (0, 0) is on the line y = 3x, so test a point off it
    2. Yes: 0 > 0 is false, so the side with (0, 0) is out
    3. Yes: 0 > 0 is true, so the side with (0, 0) is in
  7. How many points with whole-number coordinates are in the region y ≥ 2x, x > 0 and y < 6?
  8. A triangle has the solid line y = 1 along its bottom, the dashed line x = 4 on its right and the solid line y = x along its top. Which inequalities describe the region inside it?
    1. y ≥ 1, x < 4 and y ≤ x
    2. y ≥ 1, x ≤ 4 and y ≤ x
    3. x ≥ 1, y < 4 and x ≤ y
    4. y ≤ 1, x < 4 and y ≥ x
  9. Stretch: the region x ≥ 0, y ≥ 0 and x + y ≤ k has exactly 10 points with whole-number coordinates. k is a whole number. What is k?

Support: For question 4: put x = 0 and y = 0 into 3x + 2y < 12. True or false? That decides which side of the line is in.

Inequalities on graphs: exit

Draw it on paper or the board; list the points and write one sentence.

  1. Draw the region x > 0, y ≥ x − 1 and y < 3, each line dashed or solid. List every point in it with whole-number coordinates. Which of your points lie on a line, and why do they count?

Exit check

One region drawn and its points listed show who handles dashed and solid lines.

The region is x > 0, y ≥ x − 1 and y < 3. Draw it, then list every point in it with whole-number coordinates.

Inequalities on graphs: regions and points

Show your working: say whether each line is dashed or solid, and test a point to choose each side.

1. Which of the points (1, 2), (2, 4) and (0, −3) are in the region y > 3x − 2? Test each one.

2. Write the inequality for each region: (a) to the left of the solid line x = 4; (b) below the dashed line y = 2x.

3. Draw the region x + 2y ≤ 6. Is its line dashed or solid? Test (0, 0) to choose the side.

4. List every point with whole-number coordinates in the region x ≥ 1, y > 1 and x + y ≤ 5.

5. A region is inside the triangle made by the solid x-axis, the solid y-axis and the dashed line x + y = 4. Write the three inequalities that describe it.

6. The dashed lines y = 2x + 1 and y = 4 − x cross. Find where. Is that point in the region y < 2x + 1 and y > 4 − x? Say why.

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