Question

Which system of linear inequalities has the point (2, 1) in its solution set?

y less-than negative x + 3. y less-than-or-equal-to one-half x + 3 On a coordinate plane, 2 lines are shown. The first solid straight line has a positive slope and goes through (negative 4, 1) and (0, 3). Everything below the line is shaded. The second dashed straight line has a negative slope and goes through (0, 3) and (3, 0). Everything to the left of the line is shaded.
y less-than negative one-half x + 3. y less-than one-half x. On a coordinate plane, 2 lines are shown. The first solid straight line has a negative slope and goes through (0, 3) and (4, 1). Everything below the line is shaded. The second dashed straight line has a positive slope and goes through (0, 0) and (2, 1). Everything below and to the right of the line is shaded.
On a coordinate plane 2 solid straight lines are shown. The first line has a positive slope and goes through (negative 4, 1) and (0, 3). Everything below the line is shaded. The second line has a negative slope and goes through (0, 3) and (3, 0). Everything below and to the left of the line is shaded.
y less-than one-half x. y less-than-or-equal-to negative one-half x + 2 On a coordinate plane, 2 straight lines are shown. The first solid line has a negative slope and goes through (0, 2) and (4, 0). Everything below the line is shaded. The second dashed line has a positive slope and goes through (negative 4, negative 2) and (0, 0). Everything below the line is shaded.

Answer 1

To determine which system of linear inequalities has the point (2, 1) in its solution set, we need to check whether the given point satisfies the inequalities in each system.

Let's test each system:

Explanation:

y<−21​x+3 and y≤12x+3y≤21​x+3:

Substitute x = 2 and y = 1 into the first inequality:

1<−12(2)+31<−21​(2)+3

1<−1+31<−1+3

1<21<2 - True.

Now, substitute x = 2 and y = 1 into the second inequality:

1≤12(2)+31≤21​(2)+3

1≤1+31≤1+3

1≤41≤4 - True.

Since both inequalities are true for the point (2, 1), this system has the point (2, 1) in its solution set.

y<−12x+3y<−21​x+3 and y<12xy<21​x:

Substitute x = 2 and y = 1 into the first inequality:

1<−12(2)+31<−21​(2)+3

1<−1+31<−1+3

1<21<2 - True.

Now, substitute x = 2 and y = 1 into the second inequality:

1<12(2)1<21​(2)

1<11<1 - Not true.

Since the second inequality is not true for the point (2, 1), this system does not have the point (2, 1) in its solution set.

y<−12x+3y<−21​x+3 and y≤12xy≤21​x:

We already tested this system in the first option, and it does have the point (2, 1) in its solution set.

y<12xy<21​x and y≤−12x+2y≤−21​x+2:

Substitute x = 2 and y = 1 into the first inequality:

1<12(2)1<21​(2)

1<11<1 - Not true.

Now, substitute x = 2 and y = 1 into the second inequality:

1≤−12(2)+21≤−21​(2)+2

1≤−1+21≤−1+2

1≤11≤1 - True.

Since the first inequality is not true for the point (2, 1), this system does not have the point (2, 1) in its solution set.

Therefore, the system of linear inequalities that has the point (2, 1) in its solution set is:

y<−12x+3y<−21​x+3 and y≤12x+3y≤21​x+3