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2
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[tex]$y \geq-\frac{1}{3} x+2$[/tex]
[tex]$y\ \textless \ 2 x+3$[/tex]
(2,2),(3,1),(4,2)
(2,2),(3,-1),(4,1)
(2,2),(1,-2),(0,2)
(2,2),(1,2),(2,0)
Answer (1)
Substitute each point into the inequalities $y ">=-"".
Verify if both inequalities hold true for each point.
Conclude that all given points (2,2), (3,1), (4,2), (3,-1), (4,1), (1,-2), (0,2), (1,2), and (2,0) satisfy both inequalities.
The points that satisfy the inequalities are: (2,2), (3,1), (4,2), (3,-1), (4,1), (1,-2), (0,2), (1,2), (2,0).
Explanation
Understanding the Problem We are given two inequalities: $y ">=-"". We need to check which of the given points satisfy both inequalities.
Testing the First Set of Points Let's test the first set of points (2,2), (3,1), (4,2).
Checking (2,2) For (2,2): Inequality 1: $2 ">=-"". Inequality 2: 2 < 2 ( 2 ) + 3 ⇒ 2 < 4 + 3 ⇒ 2 < 7 . This is true. So, (2,2) satisfies both inequalities.
Checking (3,1) For (3,1): Inequality 1: $1 ">=-"". Inequality 2: 1 < 2 ( 3 ) + 3 ⇒ 1 < 6 + 3 ⇒ 1 < 9 . This is true. So, (3,1) satisfies both inequalities.
Checking (4,2) For (4,2): Inequality 1: $2 ">=-"". Inequality 2: 2 < 2 ( 4 ) + 3 ⇒ 2 < 8 + 3 ⇒ 2 < 11 . This is true. So, (4,2) satisfies both inequalities.
Conclusion for the First Set Therefore, (2,2), (3,1), (4,2) all satisfy both inequalities.
Testing the Second Set of Points Now let's test the second set of points (2,2), (3,-1), (4,1). We already know (2,2) satisfies both inequalities.
Checking (3,-1) For (3,-1): Inequality 1: =-"". Inequality 2:"> − 1" >= − "". I n e q u a l i t y 2 : -1 < 2(3) + 3 \Rightarrow -1 < 6 + 3 \Rightarrow -1 < 9$. This is true. So, (3,-1) satisfies both inequalities.
Checking (4,1) For (4,1): Inequality 1: $1 ">=-"". Inequality 2: 1 < 2 ( 4 ) + 3 ⇒ 1 < 8 + 3 ⇒ 1 < 11 . This is true. So, (4,1) satisfies both inequalities.
Conclusion for the Second Set Therefore, (2,2), (3,-1), (4,1) all satisfy both inequalities.
Testing the Third Set of Points Now let's test the third set of points (2,2), (1,-2), (0,2). We already know (2,2) satisfies both inequalities.
Checking (1,-2) For (1,-2): Inequality 1: =-"". Inequality 2:"> − 2" >= − "". I n e q u a l i t y 2 : -2 < 2(1) + 3 \Rightarrow -2 < 2 + 3 \Rightarrow -2 < 5$. This is true. So, (1,-2) satisfies both inequalities.
Checking (0,2) For (0,2): Inequality 1: $2 ">=-"". Inequality 2: 2 < 2 ( 0 ) + 3 ⇒ 2 < 0 + 3 ⇒ 2 < 3 . This is true. So, (0,2) satisfies both inequalities.
Conclusion for the Third Set Therefore, (2,2), (1,-2), (0,2) all satisfy both inequalities.
Testing the Fourth Set of Points Now let's test the fourth set of points (2,2), (1,2), (2,0). We already know (2,2) satisfies both inequalities.
Checking (1,2) For (1,2): Inequality 1: $2 ">=-"". Inequality 2: 2 < 2 ( 1 ) + 3 ⇒ 2 < 2 + 3 ⇒ 2 < 5 . This is true. So, (1,2) satisfies both inequalities.
Checking (2,0) For (2,0): Inequality 1: $0 ">=-"". Inequality 2: 0 < 2 ( 2 ) + 3 ⇒ 0 < 4 + 3 ⇒ 0 < 7 . This is true. So, (2,0) satisfies both inequalities.
Conclusion for the Fourth Set Therefore, (2,2), (1,2), (2,0) all satisfy both inequalities.
Final Conclusion All the given points in all sets satisfy both inequalities.
Examples
Understanding inequalities helps in various real-life situations, such as budgeting. For example, if you want to spend less than a certain amount on groceries ( y < 2 x + 3 , where x is the number of items), or if you need to maintain a minimum score on a test ( y ≥ − 3 1 x + 2 , where x is the number of questions answered), inequalities help you make informed decisions and stay within desired limits. This concept is also crucial in economics for modeling supply and demand, and in engineering for setting tolerance levels.
