Lesson Worksheet: Solving Linear–Quadratic Systems of Inequalities Mathematics

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In this worksheet, we will practice solving linear–quadratic systems of inequalities by graphing them and identifying the regions representing the solution.

Q1:

Identify the region that satisfies the following inequalities:

𝑦≥𝑥−3𝑥−4,3𝑦−𝑥<1.

  • A
  • B
  • C
  • D
  • E

Q2:

Identify the region that satisfies the following inequalities: 𝑦<−𝑥−𝑥+6,𝑦≤2𝑥+2.

  • A
  • B
  • C
  • D
  • E

Q3:

Identify the region that satisfies the following inequalities: 𝑦>(4+𝑥)(2−𝑥),𝑦+2𝑥≤4.

  • A
  • B
  • C
  • D
  • E

Q4:

Identify the region that satisfies the following inequalities:

𝑦+4≥𝑥,𝑥≥𝑦+12.

  • A
  • B
  • C
  • D
  • E

Q5:

Identify the shaded region that satisfies the three following inequalities: 𝑦≥(1−𝑥),𝑦−5>𝑥,𝑦+𝑥≤7.

  • A
  • B
  • C
  • D
  • E

Q6:

Identify the shaded region that satisfies the following three inequalities: 𝑥>1−𝑦,3𝑥≤𝑦−1,9≥−(𝑥+𝑦).

  • A
  • B
  • C
  • D
  • E

Q7:

Find the system of inequalities whose solutions are represented by the shaded area in the graph.

  • A𝑦≤−12(𝑥−2)+3,𝑦>𝑥−1
  • B𝑦≥12(𝑥−2)+3,𝑦>𝑥+1
  • C𝑦≥−12(𝑥−2)+3,𝑦<𝑥−1
  • D𝑦≥−12(𝑥−2)+3,𝑦>𝑥−1
  • E𝑦≤−12(𝑥+2)+3,𝑦>𝑥+1

Q8:

The following sketch shows three curves: 𝑦=𝑥−3𝑥−10, 𝑦=𝑥+2, and 𝑦=−2𝑥+2.

What are the inequalities that specify the shaded region?

  • A𝑦>𝑥−3𝑥−10,
    𝑦≥−𝑥+2,
    𝑦≤2𝑥+2
  • B𝑦<𝑥−3𝑥−10,
    𝑦≤𝑥+2,
    𝑦≥−2𝑥+2
  • C𝑦>𝑥−3𝑥−10,
    𝑦≥𝑥+2,
    𝑦≤−2𝑥+2
  • D𝑦>𝑥−3𝑥−10,
    𝑦≤𝑥+2,
    𝑦≥−2𝑥+2
  • E𝑦≥𝑥−3𝑥−10,
    𝑦<𝑥+2,
    𝑦>−2𝑥+2

Q9:

The parabola in the sketch has the equation 𝑦=−𝑥+3𝑥+4.

Which of the following systems of inequalities defines the shaded region?

  • A𝑦≤−𝑥+3𝑥+4,
    𝑦>2𝑥+2,
    𝑦≤−𝑥+4
  • B𝑦≤−𝑥+3𝑥+4,
    𝑦<2𝑥+2,
    𝑦≤−𝑥+4
  • C𝑦≥−𝑥+3𝑥+4,
    𝑦>2𝑥+2,
    𝑦≤−𝑥+4
  • D𝑦≤−𝑥+3𝑥+4,
    𝑦>2𝑥+2,
    𝑦≥−𝑥+4
  • E𝑦≥−𝑥+3𝑥+4,
    𝑦<2𝑥+2,
    𝑦≥−𝑥+4

Q10:

The sketch shows a parabola and two straight lines.

The parabola crosses the 𝑥-axis at the points (−6,0) and (2,0). It intersects the straight lines at the points (−14,8), (−2,−1), (−6,0), and (6,3).

Find the system of inequalities corresponding to the shaded region.

  • A𝑦<116𝑥+14𝑥−34,
    𝑦≤−34𝑥−52,
    𝑦≥14𝑥+3
  • B𝑦≤116𝑥+14𝑥−34,
    𝑦≤−34𝑥−52,
    𝑦>14𝑥+32
  • C𝑦≤116(𝑥−2)(𝑥+6),
    𝑦≤−34𝑥−52,
    𝑦>4𝑥+32
  • D𝑦≤(𝑥−2)(𝑥+6),
    𝑦≤−34𝑥−52,
    𝑦>14𝑥+32
  • E𝑦≤14𝑥+𝑥−34,
    𝑦≤−34𝑥−52,
    𝑦>14𝑥+32

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