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Sagot :
To determine which points satisfy the given system of inequalities:
[tex]$ \begin{aligned} y &\geq 5x - 3 \\ x &< 5 \end{aligned} $[/tex]
let's test each point one by one.
### Point A: (2, 8)
1. Check [tex]\( y \geq 5x - 3 \)[/tex]:
[tex]\[ 8 \geq 5(2) - 3 \implies 8 \geq 10 - 3 \implies 8 \geq 7 \quad (\text{True}) \][/tex]
2. Check [tex]\( x < 5 \)[/tex]:
[tex]\[ 2 < 5 \quad (\text{True}) \][/tex]
Since both conditions are satisfied, point A [tex]\((2, 8)\)[/tex] is a solution.
### Point B: (6, 11)
1. Check [tex]\( y \geq 5x - 3 \)[/tex]:
[tex]\[ 11 \geq 5(6) - 3 \implies 11 \geq 30 - 3 \implies 11 \geq 27 \quad (\text{False}) \][/tex]
2. Check [tex]\( x < 5 \)[/tex]:
[tex]\[ 6 < 5 \quad (\text{False}) \][/tex]
Since neither condition is satisfied, point B [tex]\((6, 11)\)[/tex] is not a solution.
### Point C: (2, 5)
1. Check [tex]\( y \geq 5x - 3 \)[/tex]:
[tex]\[ 5 \geq 5(2) - 3 \implies 5 \geq 10 - 3 \implies 5 \geq 7 \quad (\text{False}) \][/tex]
2. Check [tex]\( x < 5 \)[/tex]:
[tex]\[ 2 < 5 \quad (\text{True}) \][/tex]
Since only one condition is satisfied, point C [tex]\((2, 5)\)[/tex] is not a solution.
### Point D: (5, 30)
1. Check [tex]\( y \geq 5x - 3 \)[/tex]:
[tex]\[ 30 \geq 5(5) - 3 \implies 30 \geq 25 - 3 \implies 30 \geq 22 \quad (\text{True}) \][/tex]
2. Check [tex]\( x < 5 \)[/tex]:
[tex]\[ 5 < 5 \quad (\text{False}) \][/tex]
Since only one condition is satisfied, point D [tex]\((5, 30)\)[/tex] is not a solution.
### Point E: (4, 17)
1. Check [tex]\( y \geq 5x - 3 \)[/tex]:
[tex]\[ 17 \geq 5(4) - 3 \implies 17 \geq 20 - 3 \implies 17 \geq 17 \quad (\text{True}) \][/tex]
2. Check [tex]\( x < 5 \)[/tex]:
[tex]\[ 4 < 5 \quad (\text{True}) \][/tex]
Since both conditions are satisfied, point E [tex]\((4, 17)\)[/tex] is a solution.
### Conclusion
The points that satisfy both inequalities are:
- A. [tex]\((2, 8)\)[/tex]
- E. [tex]\((4, 17)\)[/tex]
Thus, the solutions to the system of inequalities are points [tex]\((2, 8)\)[/tex] and [tex]\((4, 17)\)[/tex].
[tex]$ \begin{aligned} y &\geq 5x - 3 \\ x &< 5 \end{aligned} $[/tex]
let's test each point one by one.
### Point A: (2, 8)
1. Check [tex]\( y \geq 5x - 3 \)[/tex]:
[tex]\[ 8 \geq 5(2) - 3 \implies 8 \geq 10 - 3 \implies 8 \geq 7 \quad (\text{True}) \][/tex]
2. Check [tex]\( x < 5 \)[/tex]:
[tex]\[ 2 < 5 \quad (\text{True}) \][/tex]
Since both conditions are satisfied, point A [tex]\((2, 8)\)[/tex] is a solution.
### Point B: (6, 11)
1. Check [tex]\( y \geq 5x - 3 \)[/tex]:
[tex]\[ 11 \geq 5(6) - 3 \implies 11 \geq 30 - 3 \implies 11 \geq 27 \quad (\text{False}) \][/tex]
2. Check [tex]\( x < 5 \)[/tex]:
[tex]\[ 6 < 5 \quad (\text{False}) \][/tex]
Since neither condition is satisfied, point B [tex]\((6, 11)\)[/tex] is not a solution.
### Point C: (2, 5)
1. Check [tex]\( y \geq 5x - 3 \)[/tex]:
[tex]\[ 5 \geq 5(2) - 3 \implies 5 \geq 10 - 3 \implies 5 \geq 7 \quad (\text{False}) \][/tex]
2. Check [tex]\( x < 5 \)[/tex]:
[tex]\[ 2 < 5 \quad (\text{True}) \][/tex]
Since only one condition is satisfied, point C [tex]\((2, 5)\)[/tex] is not a solution.
### Point D: (5, 30)
1. Check [tex]\( y \geq 5x - 3 \)[/tex]:
[tex]\[ 30 \geq 5(5) - 3 \implies 30 \geq 25 - 3 \implies 30 \geq 22 \quad (\text{True}) \][/tex]
2. Check [tex]\( x < 5 \)[/tex]:
[tex]\[ 5 < 5 \quad (\text{False}) \][/tex]
Since only one condition is satisfied, point D [tex]\((5, 30)\)[/tex] is not a solution.
### Point E: (4, 17)
1. Check [tex]\( y \geq 5x - 3 \)[/tex]:
[tex]\[ 17 \geq 5(4) - 3 \implies 17 \geq 20 - 3 \implies 17 \geq 17 \quad (\text{True}) \][/tex]
2. Check [tex]\( x < 5 \)[/tex]:
[tex]\[ 4 < 5 \quad (\text{True}) \][/tex]
Since both conditions are satisfied, point E [tex]\((4, 17)\)[/tex] is a solution.
### Conclusion
The points that satisfy both inequalities are:
- A. [tex]\((2, 8)\)[/tex]
- E. [tex]\((4, 17)\)[/tex]
Thus, the solutions to the system of inequalities are points [tex]\((2, 8)\)[/tex] and [tex]\((4, 17)\)[/tex].
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