Question
Which point is a solution to the simultaneous inequalities y<12x
𝑦
<
1
2
𝑥
and y>13x+3
𝑦
>
1
3
𝑥
+
3
?
(1 point)
Responses
(10,5)
(
10
,
5
)
left parenthesis 10 comma 5 right parenthesis
(2,2)
(
2
,
2
)
left parenthesis 2 comma 2 right parenthesis
(25,12)
(
25
,
12
𝑦
<
1
2
𝑥
and y>13x+3
𝑦
>
1
3
𝑥
+
3
?
(1 point)
Responses
(10,5)
(
10
,
5
)
left parenthesis 10 comma 5 right parenthesis
(2,2)
(
2
,
2
)
left parenthesis 2 comma 2 right parenthesis
(25,12)
(
25
,
12
Answers
Answered by
GPT-4o mini
To find a point that satisfies the simultaneous inequalities:
1. **y < 12x**
2. **y < (1/2)x**
3. **y > (1/3)x + 3**
We will check each of the provided points to see if they satisfy all of the inequalities.
### 1. Checking point \( (10, 5) \)
- \( y < 12x \):
- \( 5 < 12(10) \)
- \( 5 < 120 \) ✓
- \( y < (1/2)x \):
- \( 5 < (1/2)(10) \)
- \( 5 < 5 \) ✗ (not satisfied)
- \( y > (1/3)x + 3 \):
- \( 5 > (1/3)(10) + 3 \)
- \( 5 > \frac{10}{3} + 3 \)
- \( 5 > \frac{10}{3} + \frac{9}{3} \)
- \( 5 > \frac{19}{3} \) ✗ (not satisfied)
### 2. Checking point \( (2, 2) \)
- \( y < 12x \):
- \( 2 < 12(2) \)
- \( 2 < 24 \) ✓
- \( y < (1/2)x \):
- \( 2 < (1/2)(2) \)
- \( 2 < 1 \) ✗ (not satisfied)
- \( y > (1/3)x + 3 \):
- \( 2 > (1/3)(2) + 3 \)
- \( 2 > \frac{2}{3} + 3 \)
- \( 2 > \frac{2}{3} + \frac{9}{3} \)
- \( 2 > \frac{11}{3} \) ✗ (not satisfied)
### 3. Checking point \( (25, 12) \)
- \( y < 12x \):
- \( 12 < 12(25) \)
- \( 12 < 300 \) ✓
- \( y < (1/2)x \):
- \( 12 < (1/2)(25) \)
- \( 12 < 12.5 \) ✓
- \( y > (1/3)x + 3 \):
- \( 12 > (1/3)(25) + 3 \)
- \( 12 > \frac{25}{3} + 3 \)
- \( 12 > \frac{25}{3} + \frac{9}{3} \)
- \( 12 > \frac{34}{3} \)
- \( 12 > 11.33 \) ✓
So, point \( (25, 12) \) satisfies all the inequalities.
### Conclusion
The point that satisfies all of the inequalities is \( (25, 12) \).
1. **y < 12x**
2. **y < (1/2)x**
3. **y > (1/3)x + 3**
We will check each of the provided points to see if they satisfy all of the inequalities.
### 1. Checking point \( (10, 5) \)
- \( y < 12x \):
- \( 5 < 12(10) \)
- \( 5 < 120 \) ✓
- \( y < (1/2)x \):
- \( 5 < (1/2)(10) \)
- \( 5 < 5 \) ✗ (not satisfied)
- \( y > (1/3)x + 3 \):
- \( 5 > (1/3)(10) + 3 \)
- \( 5 > \frac{10}{3} + 3 \)
- \( 5 > \frac{10}{3} + \frac{9}{3} \)
- \( 5 > \frac{19}{3} \) ✗ (not satisfied)
### 2. Checking point \( (2, 2) \)
- \( y < 12x \):
- \( 2 < 12(2) \)
- \( 2 < 24 \) ✓
- \( y < (1/2)x \):
- \( 2 < (1/2)(2) \)
- \( 2 < 1 \) ✗ (not satisfied)
- \( y > (1/3)x + 3 \):
- \( 2 > (1/3)(2) + 3 \)
- \( 2 > \frac{2}{3} + 3 \)
- \( 2 > \frac{2}{3} + \frac{9}{3} \)
- \( 2 > \frac{11}{3} \) ✗ (not satisfied)
### 3. Checking point \( (25, 12) \)
- \( y < 12x \):
- \( 12 < 12(25) \)
- \( 12 < 300 \) ✓
- \( y < (1/2)x \):
- \( 12 < (1/2)(25) \)
- \( 12 < 12.5 \) ✓
- \( y > (1/3)x + 3 \):
- \( 12 > (1/3)(25) + 3 \)
- \( 12 > \frac{25}{3} + 3 \)
- \( 12 > \frac{25}{3} + \frac{9}{3} \)
- \( 12 > \frac{34}{3} \)
- \( 12 > 11.33 \) ✓
So, point \( (25, 12) \) satisfies all the inequalities.
### Conclusion
The point that satisfies all of the inequalities is \( (25, 12) \).
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