Which point is a solution to the following system of inequalities?
[tex]
\begin{array}{l}
Y\ \textless \ -2 x+4 \\
X \geq-2 \\
Y\ \textgreater \ -4
\end{array}
[/tex]
(-5,2)
(1,-6)
(4,1)
(1,-3)
Answer (1)
Substitute each point into the inequalities.
Check if all three inequalities are true for each point.
Point ( − 5 , 2 ) : 2 < 14 (True), − 5 ≥ − 2 (False), -4"> 2 > − 4 (True). Not a solution.
Point ( 1 , − 6 ) : − 6 < 2 (True), 1 ≥ − 2 (True), -4"> − 6 > − 4 (False). Not a solution.
Point ( 4 , 1 ) : 1 < − 4 (False), 4 ≥ − 2 (True), -4"> 1 > − 4 (True). Not a solution.
Point ( 1 , − 3 ) : − 3 < 2 (True), 1 ≥ − 2 (True), -4"> − 3 > − 4 (True). This is a solution.
The solution is ( 1 , − 3 ) .
Explanation
Analyze the problem We are given a system of inequalities:
Y < − 2 x + 4
=-2"> X " >= − 2
-4"> Y > − 4
We need to check which of the given points satisfies all three inequalities. Let's test each point.
Test point (-5, 2) Point 1: ( − 5 , 2 )
Inequality 1: 2 < − 2 ( − 5 ) + 4 ⇒ 2 < 10 + 4 ⇒ 2 < 14 . This is true.
Inequality 2: − 5 ≥ − 2 . This is false.
Inequality 3: -4"> 2 > − 4 . This is true.
Since the second inequality is not satisfied, ( − 5 , 2 ) is not a solution.
Test point (1, -6) Point 2: ( 1 , − 6 )
Inequality 1: − 6 < − 2 ( 1 ) + 4 ⇒ − 6 < − 2 + 4 ⇒ − 6 < 2 . This is true.
Inequality 2: 1 ≥ − 2 . This is true.
Inequality 3: -4"> − 6 > − 4 . This is false.
Since the third inequality is not satisfied, ( 1 , − 6 ) is not a solution.
Test point (4, 1) Point 3: ( 4 , 1 )
Inequality 1: 1 < − 2 ( 4 ) + 4 ⇒ 1 < − 8 + 4 ⇒ 1 < − 4 . This is false.
Inequality 2: 4 ≥ − 2 . This is true.
Inequality 3: -4"> 1 > − 4 . This is true.
Since the first inequality is not satisfied, ( 4 , 1 ) is not a solution.
Test point (1, -3) Point 4: ( 1 , − 3 )
Inequality 1: − 3 < − 2 ( 1 ) + 4 ⇒ − 3 < − 2 + 4 ⇒ − 3 < 2 . This is true.
Inequality 2: 1 ≥ − 2 . This is true.
Inequality 3: -4"> − 3 > − 4 . This is true.
Since all three inequalities are satisfied, ( 1 , − 3 ) is a solution.
Final Answer Therefore, the point ( 1 , − 3 ) is a solution to the system of inequalities.
Examples
Systems of inequalities are used in various real-world applications, such as linear programming, where you want to optimize a certain objective function subject to constraints. For example, a company might want to maximize its profit given constraints on resources like labor, materials, and production capacity. Each constraint can be expressed as an inequality, and the solution to the system of inequalities represents the feasible region where all constraints are satisfied. The optimal solution (e.g., maximum profit) can then be found within this feasible region.
