Question

Below are two inequalities and the graphs of their lines without the shading. By imagining where the shading should be, identify which point would satisfy BOTH inequalities.
y, is greater than, minus, x, plus, 1
y>−x+1
y, is greater than, one third, x, plus, 3
y>
3
1
​
x+3
x
y
Answer
Multiple Choice Answers
left parenthesis, minus, 8, comma, 7, right parenthesis(−8,7)
left parenthesis, minus, 6, comma, minus, 8, right parenthesis(−6,−8)
left parenthesis, 7, comma, 10, right parenthesis(7,10)
left parenthesis, 5, comma, 3, right parenthesis(5,3)

Answer 1

To determine which point satisfies both inequalities, we can evaluate each point against the inequalities y > -x + 1 and y > (1/3)x + 3.

1. **Inequality 1: y > -x + 1**
- Rearranged, this line has a slope of -1 and y-intercept of 1.

2. **Inequality 2: y > (1/3)x + 3**
- This line has a slope of 1/3 and y-intercept of 3.

Let's evaluate each point:

1. **Point (-8, 7)**:
- For y > -x + 1: 7 > -(-8) + 1 → 7 > 9 (False)
- Not a solution.

2. **Point (-6, -8)**:
- For y > -x + 1: -8 > -(-6) + 1 → -8 > 7 (False)
- Not a solution.

3. **Point (7, 10)**:
- For y > -x + 1: 10 > -7 + 1 → 10 > -6 (True)
- For y > (1/3)x + 3: 10 > (1/3)(7) + 3 → 10 > (7/3) + 3 → 10 > 2.33 + 3 → 10 > 5.33 (True)
- This point satisfies both inequalities.

4. **Point (5, 3)**:
- For y > -x + 1: 3 > -5 + 1 → 3 > -4 (True)
- For y > (1/3)x + 3: 3 > (1/3)(5) + 3 → 3 > (5/3) + 3 → 3 > 1.67 + 3 → 3 > 4.67 (False)
- Not a solution.

Based on the evaluations, **the point (7, 10)** satisfies both inequalities. The answer is:

**(7, 10)**