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9th grade · Math · Unit 18 of 22

Graphs of Equations & Inequalities

Lines, half-planes and systems

What your child practices

Writing equations and inequalities in two variables from real situations, knowing that a graph is every point that makes the equation true, and graphing inequalities as shaded half-planes with solid or dashed lines.

New York State Learning Standards learning standard: AI-A.CED.2; AI-A.REI.10; AI-A.REI.12 · creating equations and inequalities in two variables, understanding a graph as the set of all solutions, and graphing linear inequalities and systems of them as half-planes

How we explain it

Children can open this short lesson before they practice.

  1. The graph of an equation in x and y is every point that makes it true.
  2. To graph y > mx + b, draw the line y = mx + b dashed (solid for ≥ or ≤) and shade the side that works.
  3. The solutions of a system of inequalities are where the shaded half-planes overlap.

Example: Is (2, 5) a solution of y > 2x − 1? Put in x = 2: 2(2) − 1 = 3. Is 5 > 3? Yes. Answer: Yes, it is in the shaded half-plane

Graphs of Equations & Inequalities in New York

In the New York State Learning Standards, this is unit 18 of 22 in 9th grade Math, after Sequences and before Comparing Functions.

Gradelings describes a child's practice on this unit as Beginning, Approaching, Meeting or Exceeding (Four practice steps). That reflects practice, not a report-card mark: the teacher decides the mark.

Source: New York State Education Department (NYSED)

Sample questions

A taste of this unit. In the app, questions change every time and get easier or harder to match your child.

  1. Question 1

    When you graph y < 3x − 3, is the boundary line solid or dashed?

    • dashed: points on the line are not solutions
    • there is no boundary line
    • solid: points on the line are solutions

  2. Question 2

    Which point is on the graph of x + 2y = −4?

    • (0, −1)
    • (−1, −1)
    • (−2, 0)
    • (0, −2)

  3. Question 3

    Which point is a solution of both y > 2x and y < 8?

    • (−1, −2)
    • (−1, 3)
    • (2, 10)
    • (−1, −3)

  4. Question 4

    Which point is a solution of y ≥ 3x?

    • (0, −4)
    • (−2, −5)
    • (−1, −4)
    • (−3, −12)

  5. Question 5

    When you graph y ≥ x − 3, is the boundary line solid or dashed?

    • there is no boundary line
    • solid: points on the line are solutions
    • dashed: points on the line are not solutions

  6. Question 6

    Is (−1, 0) on the graph of y = 2x + 2?

    • Yes: it makes the equation true
    • No: it does not make the equation true

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