Graphical Inequalities and Regions in the Plane · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Shading the side that passes the test

Mathematics · Algebra · ages 15-16
Name ______________________   Date ____________

  1. The region y < 2x + 1 is shaded. Should its boundary line be dashed or solid?

    • Dashed, because points on the line are not included
    • Solid, because points on the line are not included
    • Dashed, because points on the line are included
    • Solid, because points on the line are included

  2. The region y less than 2x plus 1 is shaded. Should its boundary line be dashed or solid?

    • Dashed, since points on the line are not included
    • Solid, since points on the line are not included
    • Solid, since points on the line are included

  3. What does a solid boundary line tell you about a region?

    • Points on the line fail the test
    • Points on the line pass the test
    • The line was drawn by mistake

  4. The point (0, 0) lies in the region y less than 2x plus 1.

    Circle one:   True   False

  5. A region includes all points on or below the line y = 4 - x. Which inequality describes it?

    • y <= 4 - x
    • y < 4 - x
    • y >= 4 - x
    • y > 4 - x

  6. Sofia says the point (2, 5) lies in the region y less than 2x plus 1.

    Circle one:   True   False

  7. A region holds x at least 0, y at least 0 and x plus y at most 3. How many integer points lie inside it?

    Answer: ______________

  8. A region is defined by x >= 0, y >= 0 and x + y <= 3. How many points with integer coordinates lie inside it?

    Answer: ______________

  9. A region holds 0 at most x at most 3 and 0 at most y at most 2. How many integer points lie inside it?

    Answer: ______________

  10. A region is defined by 0 <= x <= 3 and 0 <= y <= 2. How many points with integer coordinates lie inside it?

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Shading the side that passes the test W1-mt_sKerzcAcNy-s1

  1. Dashed, because points on the line are not included · A strict inequality excludes its boundary, and excluded boundaries are drawn dashed.
  2. Dashed, since points on the line are not included · A strict sign excludes its boundary, and excluded boundaries are dashed.
  3. Points on the line pass the test · Solid means inclusive, so the boundary itself counts as inside.
  4. True · Putting in zero gives 0 less than 1, which is true.
  5. y <= 4 - x · Below the line means y is smaller, and including the line means or equal to, so y <= 4 - x.
  6. False · Putting in the point gives 5 less than 5, which is false.
  7. 10 · Listing by x gives 4 plus 3 plus 2 plus 1, which is 10 points.
  8. 10 · Listing by x gives 4 + 3 + 2 + 1 = 10 integer points inside the triangle.
  9. 12 · Four integer x options times three integer y options gives 12 pairs.
  10. 12 · x has 4 integer options and y has 3, and every pair fits, so 4 x 3 = 12 points.
Worksheet · LightMySky