Write a linear inequality in two variables that has the following properties:
- $(2,-1),(2,3)$, and $(3,1)$ are not solutions.
- $(0,-3),(-2,1)$, and $(1,-5)$ are solutions.
Answer (1)
Substitute the given points into the inequality a x + b y < c to find constraints on a , b , and c .
Choose values for a , b , and c that satisfy the constraints.
Check if the chosen values satisfy all the inequalities.
The linear inequality that satisfies the given conditions is x + y < 0 .
Explanation
Problem Analysis We are looking for a linear inequality in two variables, x and y , that satisfies two conditions:
The points ( 2 , − 1 ) , ( 2 , 3 ) , and ( 3 , 1 ) are NOT solutions.
The points ( 0 , − 3 ) , ( − 2 , 1 ) , and ( 1 , − 5 ) ARE solutions.
Let's assume the inequality has the form a x + b y < c . We will substitute the given points into this inequality to find constraints on a , b , and c .
Substituting Points First, let's substitute the points that are NOT solutions into the inequality a x + b y < c . Since these points are not solutions, the inequality must not hold true for these points. This means that for these points, a x + b y ≥ c .
For ( 2 , − 1 ) : 2 a − b ≥ c
For ( 2 , 3 ) : 2 a + 3 b ≥ c
For ( 3 , 1 ) : 3 a + b ≥ c
Now, let's substitute the points that ARE solutions into the inequality a x + b y < c . This means that for these points, the inequality must hold true.
For ( 0 , − 3 ) : 0 a − 3 b < c ⇒ − 3 b < c
For ( − 2 , 1 ) : − 2 a + b < c
For ( 1 , − 5 ) : a − 5 b < c
Finding Values for a, b, and c Let's try to find a set of values for a , b , and c that satisfy these inequalities. We can start by choosing a value for b and see if we can find corresponding values for a and c .
Let's choose b = 1 . Then we have:
− 3 < c
2 a − 1 ≥ c
2 a + 3 ≥ c
3 a + 1 ≥ c
− 2 a + 1 < c
a − 5 < c
From − 3 < c , let's choose c = 0 . Then we have:
2 a − 1 ≥ 0 ⇒ a ≥ 2 1
2 a + 3 ≥ 0 ⇒ a ≥ − 2 3
3 a + 1 ≥ 0 ⇒ a ≥ − 3 1
\frac{1}{2}"> − 2 a + 1 < 0 ⇒ a > 2 1
a − 5 < 0 ⇒ a < 5
So, we need \frac{1}{2}"> a > 2 1 and a < 5 . Let's choose a = 1 .
Checking the Inequality Now we have a = 1 , b = 1 , and c = 0 . Let's check if these values satisfy all the inequalities:
2 ( 1 ) − 1 ≥ 0 ⇒ 1 ≥ 0 (True)
2 ( 1 ) + 3 ( 1 ) ≥ 0 ⇒ 5 ≥ 0 (True)
3 ( 1 ) + 1 ≥ 0 ⇒ 4 ≥ 0 (True)
− 3 ( 1 ) < 0 ⇒ − 3 < 0 (True)
− 2 ( 1 ) + 1 < 0 ⇒ − 1 < 0 (True)
1 − 5 ( 1 ) < 0 ⇒ − 4 < 0 (True)
So, the inequality x + y < 0 satisfies the given conditions.
Final Answer Therefore, a linear inequality that satisfies the given conditions is x + y < 0 .
Examples
Linear inequalities are used in various real-world scenarios, such as resource allocation. For example, a company might use a linear inequality to determine the optimal combination of labor and capital to minimize costs while meeting production targets. The inequality would represent the constraint on resources, and the solution would provide the feasible region of labor and capital combinations.
