Is $(0,-3)$ a solution to the system of inequalities below:
[tex]
\begin{array}{l}
y\ \textless \ x+5 \
y \geq 2 x-4
\end{array}
[/tex]
Type yes or type no for your answer.
Answer (2)
The point (0, -3) satisfies both inequalities in the system, making it a solution. Therefore, the answer is yes. Both inequalities are true when we substitute x = 0 and y = -3.
;
Substitute x = 0 and y = − 3 into the first inequality y < x + 5 , which gives − 3 < 0 + 5 , simplifying to − 3 < 5 . This is true.
Substitute x = 0 and y = − 3 into the second inequality y ≥ 2 x − 4 , which gives − 3 ≥ 2 ( 0 ) − 4 , simplifying to − 3 ≥ − 4 . This is true.
Since both inequalities are true, the point ( 0 , − 3 ) is a solution to the system of inequalities.
Therefore, the answer is yes .
Explanation
Problem Analysis We are given the system of inequalities:
y < x + 5 = 2x - 4"> y " >= 2 x − 4
We need to check if the point ( 0 , − 3 ) is a solution to this system. This means we need to substitute x = 0 and y = − 3 into both inequalities and see if they are both true.
Checking the First Inequality First, let's substitute x = 0 and y = − 3 into the first inequality:
y < x + 5 − 3 < 0 + 5 − 3 < 5
This inequality is true.
Checking the Second Inequality Now, let's substitute x = 0 and y = − 3 into the second inequality:
y ≥ 2 x − 4 − 3 ≥ 2 ( 0 ) − 4 − 3 ≥ 0 − 4 − 3 ≥ − 4
This inequality is also true.
Conclusion Since both inequalities are true when we substitute x = 0 and y = − 3 , the point ( 0 , − 3 ) is a solution to the system of inequalities.
Examples
Systems of inequalities are used in various real-world applications, such as optimizing resource allocation. For example, a company might use a system of inequalities to determine the optimal production levels of two different products, given constraints on available resources like labor and materials. By graphing the inequalities, the company can identify the feasible region, representing all possible production levels that satisfy the constraints. The company can then use optimization techniques to find the production levels that maximize profit within the feasible region. This approach helps businesses make informed decisions about resource allocation and production planning.
