To determine which ordered pair makes both inequalities true, we will test each pair against the conditions given.
The inequalities are:
[tex]\[ y < 3x - 1 \][/tex]
[tex]\[ y \geq -x + 4 \][/tex]
Let's check each of the given points one by one.
1. Point (4, 0):
- First inequality: [tex]\( y < 3x - 1 \)[/tex]
[tex]\[ 0 < 3(4) - 1 \][/tex]
[tex]\[ 0 < 12 - 1 \][/tex]
[tex]\[ 0 < 11 \][/tex]
This is true.
- Second inequality: [tex]\( y \geq -x + 4 \)[/tex]
[tex]\[ 0 \geq -4 + 4 \][/tex]
[tex]\[ 0 \geq 0 \][/tex]
This is true.
So, the point (4, 0) satisfies both inequalities.
2. Point (1, 2):
- First inequality: [tex]\( y < 3x - 1 \)[/tex]
[tex]\[ 2 < 3(1) - 1 \][/tex]
[tex]\[ 2 < 3 - 1 \][/tex]
[tex]\[ 2 < 2 \][/tex]
This is false.
Therefore, the point (1, 2) does not satisfy the first inequality, and we do not need to check the second one.
3. Point (0, 4):
- First inequality: [tex]\( y < 3x - 1 \)[/tex]
[tex]\[ 4 < 3(0) - 1 \][/tex]
[tex]\[ 4 < 0 - 1 \][/tex]
[tex]\[ 4 < -1 \][/tex]
This is false.
Therefore, the point (0, 4) does not satisfy the first inequality, and we do not need to check the second one.
4. Point (2, 1):
- First inequality: [tex]\( y < 3x - 1 \)[/tex]
[tex]\[ 1 < 3(2) - 1 \][/tex]
[tex]\[ 1 < 6 - 1 \][/tex]
[tex]\[ 1 < 5 \][/tex]
This is true.
- Second inequality: [tex]\( y \geq -x + 4 \)[/tex]
[tex]\[ 1 \geq -2 + 4 \][/tex]
[tex]\[ 1 \geq 2 \][/tex]
This is false.
Therefore, the point (2, 1) does not satisfy the second inequality.
After evaluating all given points, the ordered pair that makes both inequalities true is:
[tex]\[ \boxed{(4, 0)} \][/tex]
