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 less than, minus, x, minus, 2
y<−x−2
y, is less than, start fraction, 2, divided by, 3, end fraction, x, plus, 6
y<
3
2
x+6
x
y
Answer
Multiple Choice Answers
left parenthesis, 4, comma, 5, right parenthesis(4,5)
left parenthesis, minus, 4, comma, 5, right parenthesis(−4,5)
left parenthesis, minus, 3, comma, minus, 1, right parenthesis(−3,−1)
left parenthesis, minus, 9, comma, 6, right parenthesis(−9,6)
y, is less than, minus, x, minus, 2
y<−x−2
y, is less than, start fraction, 2, divided by, 3, end fraction, x, plus, 6
y<
3
2
x+6
x
y
Answer
Multiple Choice Answers
left parenthesis, 4, comma, 5, right parenthesis(4,5)
left parenthesis, minus, 4, comma, 5, right parenthesis(−4,5)
left parenthesis, minus, 3, comma, minus, 1, right parenthesis(−3,−1)
left parenthesis, minus, 9, comma, 6, right parenthesis(−9,6)
Answers
GPT-4o mini
To determine which point satisfies both inequalities, we first need to analyze the inequalities and their respective lines.
1. **Inequality 1:** \( y < -x - 2 \)
- Rearranging gives the line \( y = -x - 2 \). The line has a slope of -1 and a y-intercept of -2. The region below this line (shaded area for the inequality) is where \( y \) values are less than the line.
2. **Inequality 2:** \( y < \frac{2}{3}x + 6 \)
- Rearranging gives the line \( y = \frac{2}{3}x + 6 \). The slope is \( \frac{2}{3} \) and y-intercept is 6. The region below this line is where \( y \) values are less than the line.
Next, we can analyze the choice points to see which one satisfies both inequalities:
1. Point **(4, 5)**:
- For \( y < -x - 2 \):
- \( 5 < -4 - 2 \)
- \( 5 < -6 \) (False)
- For \( y < \frac{2}{3}x + 6 \):
- \( 5 < \frac{2}{3}(4) + 6 \)
- \( 5 < \frac{8}{3} + 6 = \frac{8}{3} + \frac{18}{3} = \frac{26}{3} \) (True)
2. Point **(-4, 5)**:
- For \( y < -x - 2 \):
- \( 5 < -(-4) - 2 \)
- \( 5 < 4 - 2 = 2 \) (False)
- For \( y < \frac{2}{3}x + 6 \):
- \( 5 < \frac{2}{3}(-4) + 6 \)
- \( 5 < -\frac{8}{3} + 6 = -\frac{8}{3} + \frac{18}{3} = \frac{10}{3} \) (False)
3. Point **(-3, -1)**:
- For \( y < -x - 2 \):
- \( -1 < -(-3) - 2 \)
- \( -1 < 3 - 2 = 1 \) (True)
- For \( y < \frac{2}{3}x + 6 \):
- \( -1 < \frac{2}{3}(-3) + 6 \)
- \( -1 < -2 + 6 = 4 \) (True)
4. Point **(-9, 6)**:
- For \( y < -x - 2 \):
- \( 6 < -(-9) - 2 \)
- \( 6 < 9 - 2 = 7 \) (True)
- For \( y < \frac{2}{3}x + 6 \):
- \( 6 < \frac{2}{3}(-9) + 6 \)
- \( 6 < -6 + 6 = 0 \) (False)
After evaluating all points:
- Only **(-3, -1)** satisfies both inequalities:
1. \( -1 < 1 \) (True)
2. \( -1 < 4 \) (True)
Thus, the point which satisfies both inequalities is ***(-3, -1)***.
1. **Inequality 1:** \( y < -x - 2 \)
- Rearranging gives the line \( y = -x - 2 \). The line has a slope of -1 and a y-intercept of -2. The region below this line (shaded area for the inequality) is where \( y \) values are less than the line.
2. **Inequality 2:** \( y < \frac{2}{3}x + 6 \)
- Rearranging gives the line \( y = \frac{2}{3}x + 6 \). The slope is \( \frac{2}{3} \) and y-intercept is 6. The region below this line is where \( y \) values are less than the line.
Next, we can analyze the choice points to see which one satisfies both inequalities:
1. Point **(4, 5)**:
- For \( y < -x - 2 \):
- \( 5 < -4 - 2 \)
- \( 5 < -6 \) (False)
- For \( y < \frac{2}{3}x + 6 \):
- \( 5 < \frac{2}{3}(4) + 6 \)
- \( 5 < \frac{8}{3} + 6 = \frac{8}{3} + \frac{18}{3} = \frac{26}{3} \) (True)
2. Point **(-4, 5)**:
- For \( y < -x - 2 \):
- \( 5 < -(-4) - 2 \)
- \( 5 < 4 - 2 = 2 \) (False)
- For \( y < \frac{2}{3}x + 6 \):
- \( 5 < \frac{2}{3}(-4) + 6 \)
- \( 5 < -\frac{8}{3} + 6 = -\frac{8}{3} + \frac{18}{3} = \frac{10}{3} \) (False)
3. Point **(-3, -1)**:
- For \( y < -x - 2 \):
- \( -1 < -(-3) - 2 \)
- \( -1 < 3 - 2 = 1 \) (True)
- For \( y < \frac{2}{3}x + 6 \):
- \( -1 < \frac{2}{3}(-3) + 6 \)
- \( -1 < -2 + 6 = 4 \) (True)
4. Point **(-9, 6)**:
- For \( y < -x - 2 \):
- \( 6 < -(-9) - 2 \)
- \( 6 < 9 - 2 = 7 \) (True)
- For \( y < \frac{2}{3}x + 6 \):
- \( 6 < \frac{2}{3}(-9) + 6 \)
- \( 6 < -6 + 6 = 0 \) (False)
After evaluating all points:
- Only **(-3, -1)** satisfies both inequalities:
1. \( -1 < 1 \) (True)
2. \( -1 < 4 \) (True)
Thus, the point which satisfies both inequalities is ***(-3, -1)***.