# Finding Systems of Inequalities That Describe Triangular and Rectangular Regions

### New York State Common Core Math Geometry, Module 4, Lesson 2

Student Outcomes

Students describe rectangles (with edges parallel to the axes) and triangles in the coordinate plane by means of inequalities. For example, the rectangle in the coordinate plane with lower left vertex (1,2) and upper right vertex (10,15) is {(x,y) l 1 < x < 10 & 2 < y < 15} , the triangle with vertices at (0,0), (1,3), and (2,1) is {(x,y) l x/2 < y < 3x & y < -2x + 5}.

Finding Systems of Inequalities That Describe Triangular and Rectangular Regions

Classwork

Opening Exercise

Graph each system of inequalities.
a. {𝑦 ≥ 1
𝑥 ≤ 5
i. Is (1,2) a solution? Explain.
ii. Is (1,1) a solution? Explain.
iii. The region is the intersection of how many half-planes? Explain how you know.

b. {𝑦 < 2𝑥 +1
𝑦 ≥ −3𝑥 −2
i. Is (−2,4) in the solution set?
ii. Is (1,3) in the solution set?
iii. The region is the intersection of how many half-planes? Explain how you know.

Exercises 1–3

1. Given the region shown to the right:
a. Name three points in the interior of the region.
b. Name three points on the boundary.
c. Describe the coordinates of the points in the region.
d. Write the inequality describing the 𝑥-values.
e. Write the inequality describing the 𝑦-values.
f. Write this as a system of equations.
g. Will the lines 𝑥 = 4 and 𝑦 = 1 pass through the region? Draw them.
2. Given the region that continues unbound to the right as shown to the right:
a. Name three points in the region.
b. Describe in words the points in the region.
c. Write the system of inequalities that describe the region.
d. Name a horizontal line that passes through the region.
3. Given the region that continues down without bound as shown to the right:
a. Describe the region in words.
b. Write the system of inequalities that describe the region.
c. Name a vertical line that passes through the region.

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