What ordered pair is the solution to the following system of linear inequalities?
[tex]\begin{array}{l}
4 x-2 y \leq 4 \\
4 y\ \textless \ 2 x+5
\end{array}[/tex]
A. (2,1)
B. [tex]\left(1, \frac{1}{2}\right)[/tex]
C. [tex]\left(\frac{1}{2},-2\right)[/tex]
D. [tex]\left(\frac{3}{4},-1\right)[/tex]
Answer (1)
Substitute each ordered pair into the inequalities.
Check if both inequalities are true for the ordered pair.
If both inequalities are true, the ordered pair is a solution.
The ordered pair ( 1 , 2 1 ) satisfies both inequalities, so the solution is ( 1 , 2 1 ) .
Explanation
Problem Analysis We are given a system of two linear inequalities and four possible solutions. We need to check which of the given ordered pairs satisfies both inequalities.
Testing (2, 1) Let's test the first ordered pair ( 2 , 1 ) in the inequalities:
Inequality 1: 4 x − 2 y ≤ 4 Substitute x = 2 and y = 1 :
4 ( 2 ) − 2 ( 1 ) ≤ 4 8 − 2 ≤ 4 6 ≤ 4 (False)
Since the first inequality is false for ( 2 , 1 ) , we don't need to check the second inequality. ( 2 , 1 ) is not a solution.
Testing (1, 1/2) Now let's test the second ordered pair ( 1 , 2 1 ) in the inequalities:
Inequality 1: 4 x − 2 y ≤ 4 Substitute x = 1 and y = 2 1 :
4 ( 1 ) − 2 ( 2 1 ) ≤ 4 4 − 1 ≤ 4 3 ≤ 4 (True)
Inequality 2: 4 y < 2 x + 5 Substitute x = 1 and y = 2 1 :
4 ( 2 1 ) < 2 ( 1 ) + 5 2 < 2 + 5 2 < 7 (True)
Since both inequalities are true for ( 1 , 2 1 ) , this ordered pair is a solution.
Final Answer Since we found a solution, we don't need to check the other ordered pairs.
Examples
Understanding systems of inequalities helps in various real-world scenarios, such as optimizing resource allocation within constraints. For instance, a company might use inequalities to determine the optimal production levels of two products, given limitations on resources like labor and materials. By identifying the feasible region that satisfies all constraints, the company can make informed decisions to maximize profit while adhering to resource limitations. This approach ensures efficient use of resources and optimal outcomes.
