To determine which point is a solution to the inequality [tex]\( y \geq 2x - 1 \)[/tex], we need to substitute each point into the inequality and see if the inequality holds true.
Let's check each option step-by-step:
### Option A: [tex]\((4, 2)\)[/tex]
1. Calculate [tex]\( 2x - 1 \)[/tex]:
[tex]\[
2 \times 4 - 1 = 8 - 1 = 7
\][/tex]
2. Compare [tex]\( y \)[/tex] with [tex]\( 2x - 1 \)[/tex]:
[tex]\[
2 \geq 7 \quad (\text{False})
\][/tex]
Point [tex]\((4, 2)\)[/tex] does not satisfy the inequality.
### Option B: [tex]\((0, -10)\)[/tex]
1. Calculate [tex]\( 2x - 1 \)[/tex]:
[tex]\[
2 \times 0 - 1 = 0 - 1 = -1
\][/tex]
2. Compare [tex]\( y \)[/tex] with [tex]\( 2x - 1 \)[/tex]:
[tex]\[
-10 \geq -1 \quad (\text{False})
\][/tex]
Point [tex]\((0, -10)\)[/tex] does not satisfy the inequality.
### Option C: [tex]\((0, 2)\)[/tex]
1. Calculate [tex]\( 2x - 1 \)[/tex]:
[tex]\[
2 \times 0 - 1 = 0 - 1 = -1
\][/tex]
2. Compare [tex]\( y \)[/tex] with [tex]\( 2x - 1 \)[/tex]:
[tex]\[
2 \geq -1 \quad (\text{True})
\][/tex]
Point [tex]\((0, 2)\)[/tex] does satisfy the inequality.
### Option D: [tex]\((4, 1)\)[/tex]
1. Calculate [tex]\( 2x - 1 \)[/tex]:
[tex]\[
2 \times 4 - 1 = 8 - 1 = 7
\][/tex]
2. Compare [tex]\( y \)[/tex] with [tex]\( 2x - 1 \)[/tex]:
[tex]\[
1 \geq 7 \quad (\text{False})
\][/tex]
Point [tex]\((4, 1)\)[/tex] does not satisfy the inequality.
### Conclusion
Only point [tex]\( \boxed{C} \, (0, 2) \)[/tex] is a solution to the inequality [tex]\( y \geq 2x - 1 \)[/tex].
