Question
Which system of linear inequalities has the point (3, –2) in its solution set?
y less-than negative 3. y less-than-or-equal-to two-thirds x minus 4 On a coordinate plane, 2 straight lines are shown. The first dashed line is horizontal to the y-axis at y = negative 3. Everything below the line is shaded. The second solid line has a positive slope and goes through (0, negative 4) and (3, negative 2). Everything below the line is shaded.
y greater-than negative 3. y greater-than-or-equal-to two-thirds x minus 4 On a coordinate plane, 2 straight lines are shown. The first dashed line is horizontal to the y-axis at y = negative 3. Everything above the line is shaded. The second solid line has a positive slope and goes through (0, negative 4) and (3, negative 2). Everything above the line is shaded.
y less-than negative 3. y greater-than-or-equal-to two-thirds x minus 4 On a coordinate plane, 2 straight lines are shown. The first dashed line is horizontal to the y-axis at y = negative 3. Everything below the line is shaded. The second solid line has a positive slope and goes through (0, negative 4) and (3, negative 2). Everything above the line is shaded.
y greater-than negative 2. y less-than-or-equal-to two-thirds x minus 4 On a coordinate plane, 2 straight lines are shown. The first dashed line is horizontal at y = negative 2. Everything above the line is shaded. The second solid line has a positive slope and goes through (0, negative 4) and (3, negative 2). Everything below the line is shaded.
y less-than negative 3. y less-than-or-equal-to two-thirds x minus 4 On a coordinate plane, 2 straight lines are shown. The first dashed line is horizontal to the y-axis at y = negative 3. Everything below the line is shaded. The second solid line has a positive slope and goes through (0, negative 4) and (3, negative 2). Everything below the line is shaded.
y greater-than negative 3. y greater-than-or-equal-to two-thirds x minus 4 On a coordinate plane, 2 straight lines are shown. The first dashed line is horizontal to the y-axis at y = negative 3. Everything above the line is shaded. The second solid line has a positive slope and goes through (0, negative 4) and (3, negative 2). Everything above the line is shaded.
y less-than negative 3. y greater-than-or-equal-to two-thirds x minus 4 On a coordinate plane, 2 straight lines are shown. The first dashed line is horizontal to the y-axis at y = negative 3. Everything below the line is shaded. The second solid line has a positive slope and goes through (0, negative 4) and (3, negative 2). Everything above the line is shaded.
y greater-than negative 2. y less-than-or-equal-to two-thirds x minus 4 On a coordinate plane, 2 straight lines are shown. The first dashed line is horizontal at y = negative 2. Everything above the line is shaded. The second solid line has a positive slope and goes through (0, negative 4) and (3, negative 2). Everything below the line is shaded.
Answers
GPT-4o mini
To determine which system of linear inequalities has the point (3, -2) in its solution set, we will check the inequalities against the point (3, -2).
1. **First option:**
- \( y < -3 \):
- Checking with the point (3, -2): -2 is not less than -3. (False)
- \( y \leq \frac{2}{3}x - 4 \):
- Substitute \( x = 3 \):
\( y \leq \frac{2}{3}(3) - 4 \)
\( y \leq 2 - 4 \)
\( y \leq -2 \)
- Since -2 is equal to -2 (true).
- The first option is **not valid** because the first inequality is false.
2. **Second option:**
- \( y > -3 \):
- Checking with the point (3, -2): -2 is greater than -3. (True)
- \( y \geq \frac{2}{3}x - 4 \):
- Substitute \( x = 3 \):
\( y \geq \frac{2}{3}(3) - 4 \)
\( y \geq 2 - 4 \)
\( y \geq -2 \)
- Since -2 is equal to -2 (true).
- The second option is **valid**.
3. **Third option:**
- \( y < -3 \):
- Checking with point (3, -2): -2 is not less than -3. (False)
- \( y \geq \frac{2}{3}x - 4 \):
- Substitute \( x = 3 \):
\( y \geq \frac{2}{3}(3) - 4 \)
\( y \geq -2 \)
- This part is true.
- The third option is **not valid** because the first inequality is false.
4. **Fourth option:**
- \( y > -2 \):
- Checking with the point (3, -2): -2 is not greater than -2. (False)
- \( y \leq \frac{2}{3}x - 4 \):
- Substitute \( x = 3 \):
\( y \leq \frac{2}{3}(3) - 4 \)
\( y \leq -2 \)
- Since -2 is equal to -2 (true).
- The fourth option is **not valid** because the first inequality is false.
After evaluating all four options, the second option is the only one where both inequalities are true for the point \( (3, -2) \):
**y > -3 and y ≥ (2/3)x - 4**.
1. **First option:**
- \( y < -3 \):
- Checking with the point (3, -2): -2 is not less than -3. (False)
- \( y \leq \frac{2}{3}x - 4 \):
- Substitute \( x = 3 \):
\( y \leq \frac{2}{3}(3) - 4 \)
\( y \leq 2 - 4 \)
\( y \leq -2 \)
- Since -2 is equal to -2 (true).
- The first option is **not valid** because the first inequality is false.
2. **Second option:**
- \( y > -3 \):
- Checking with the point (3, -2): -2 is greater than -3. (True)
- \( y \geq \frac{2}{3}x - 4 \):
- Substitute \( x = 3 \):
\( y \geq \frac{2}{3}(3) - 4 \)
\( y \geq 2 - 4 \)
\( y \geq -2 \)
- Since -2 is equal to -2 (true).
- The second option is **valid**.
3. **Third option:**
- \( y < -3 \):
- Checking with point (3, -2): -2 is not less than -3. (False)
- \( y \geq \frac{2}{3}x - 4 \):
- Substitute \( x = 3 \):
\( y \geq \frac{2}{3}(3) - 4 \)
\( y \geq -2 \)
- This part is true.
- The third option is **not valid** because the first inequality is false.
4. **Fourth option:**
- \( y > -2 \):
- Checking with the point (3, -2): -2 is not greater than -2. (False)
- \( y \leq \frac{2}{3}x - 4 \):
- Substitute \( x = 3 \):
\( y \leq \frac{2}{3}(3) - 4 \)
\( y \leq -2 \)
- Since -2 is equal to -2 (true).
- The fourth option is **not valid** because the first inequality is false.
After evaluating all four options, the second option is the only one where both inequalities are true for the point \( (3, -2) \):
**y > -3 and y ≥ (2/3)x - 4**.