Pregunta
Which point would be a solution to the system of linear inequalities
shown below?
\[ y-\frac{2}{3} x-7 \]
Answer
\( \begin{array}{l}(-6,-7) \\ (-6,10) \\ (3,0) \\ (-9,0)\end{array} \)
Ask by Donnelly Gibson. in the United States
Feb 25,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
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The point (3, 0) is the solution to the system of inequalities.
Solución
To solve the system, we substitute each point into the inequalities and check:
System:
(1) y < x + 1
(2) y > (-2/3)x – 7
Check each candidate:
1. For (–6, –7):
Inequality (1): –7 < (–6 + 1) → –7 < –5 (true)
Inequality (2): –7 > (–2/3)(–6) – 7 = 4 – 7 = –3 → –7 > –3 (false)
2. For (–6, 10):
Inequality (1): 10 < (–6 + 1) → 10 < –5 (false)
3. For (3, 0):
Inequality (1): 0 < (3 + 1) → 0 < 4 (true)
Inequality (2): 0 > (–2/3)(3) – 7 = –2 – 7 = –9 → 0 > –9 (true)
4. For (–9, 0):
Inequality (1): 0 < (–9 + 1) → 0 < –8 (false)
Only point (3, 0) satisfies both inequalities.
Thus, the solution is (3, 0).
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Simplifique esta solución Extra Insights
To determine which point satisfies the system of inequalities, we can substitute the coordinates of each point into the inequalities. For instance, the point \((-6,10)\) leads to \(10 < -6 + 1\) (which is false) and \(10 > -\frac{2}{3}(-6) - 7\) (which is true). So this point isn’t the solution. Now, let’s check the final point \((-9,0)\). For this point, we find \(0 < -9 + 1\) (which is false) and \(0 > -\frac{2}{3}(-9) - 7\) (which is true). Testing each point diligently will lead to the right solution!
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