En anglais

Checking points and describing inequality regions

Mathematics · Algebra · ages 15-16

Name ______________________ Date ____________

To decide which side of y = 3x to shade for y > 3x, which point can be used as a test point?
- (0, 0)
- (1, 3)
- (0, 2)
- (-1, -3)
The region y less than 2x plus 1 is shaded. Should its boundary line be dashed or solid?
- Dashed; the boundary is excluded
- Solid; every boundary point is excluded
- Solid; every boundary point is included
For a single linear inequality, what does a solid boundary line tell you?
- All points on it fail that inequality
- All points on it satisfy that inequality
- All points on it satisfy every inequality
The point (0, 0) lies in the region y less than 2x plus 1.
Circle one: True False
A region is right of a solid line x = -1 and above a dashed line y = 2. Which pair of inequalities describes it?
- x > -1 and y >= 2
- x >= -1 and y > 2
- x <= -1 and y > 2
- x >= -1 and y < 2
Sofia says the point (2, 5) lies in the region y less than 2x plus 1.
Circle one: True False
How many integer-coordinate points satisfy x >= 0, y >= 0 and x + y <= 3? Count allowed boundary points too.
Answer: ______________
How many integer-coordinate points satisfy x >= 0, y > 0 and x + y < 4?
Answer: ______________
How many integer-coordinate points satisfy 0 <= x <= 3 and 0 <= y <= 2? Count allowed boundary points too.
Answer: ______________
How many integer-coordinate points satisfy 0 <= x <= 2, y >= x and y <= 4 - x?
Answer: ______________

Answer key

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

(0, 2) · A test point must be off the boundary. Only (0, 2) is off y = 3x. Its test is 2 > 0, so shade its side.
Dashed; the boundary is excluded · A strict sign excludes its boundary, and excluded boundaries are dashed.
All points on it satisfy that inequality · A solid boundary includes equality, so its points satisfy that inequality. Other inequalities in a system still need checking.
True · Putting in zero gives 0 less than 1, which is true.
x >= -1 and y > 2 · Right of x = -1 gives larger x-values, with equality for the solid boundary. Above y = 2 gives larger y-values, without equality for the dashed boundary.
False · Putting in the point gives 5 less than 5, which is false.
10 · Including the allowed boundary points, the columns contain 4, 3, 2 and 1 points. Their sum is 10.
6 · Exclude y = 0 and x + y = 4. The nonempty columns contain 3, 2 and 1 integer points, totaling 6.
12 · Four integer x options times three integer y options gives 12 pairs.
9 · For x = 0, 1 and 2, the integer y counts are 5, 3 and 1. Their sum is 9.