Which ordered pairs make both inequalities true?
[tex]
\begin{array}{l}
y\ \textless \ 5 x+2 \\
y \geq \frac{1}{2} x+1
\end{array}
[/tex]
Answer (1)
Test the ordered pair (0, 1) in both inequalities and confirm that both inequalities are true.
Test the ordered pair (1, 6) in both inequalities and confirm that both inequalities are true.
Conclude that both (0, 1) and (1, 6) are solutions to the system of inequalities.
The ordered pairs that satisfy both inequalities are ( 0 , 1 ) , ( 1 , 6 ) .
Explanation
Analyze the problem and data We are given two inequalities: y < 5 x + 2 = "$\frac{1}{2}x + 1"> y ">= "$\frac{1}{2}x + 1 We need to find two ordered pairs ( x , y ) that satisfy both inequalities. To do this, we will test several ordered pairs to see if they satisfy both inequalities.
Test the ordered pair (0, 1) Let's test the ordered pair ( 0 , 1 ) .
For the first inequality, we have: 1 < 5 ( 0 ) + 2 1 < 2 This is true. For the second inequality, we have: = "$\frac{1}{2}(0) + 1"> 1 ">= "$\frac{1}{2}(0) + 1 = "1"> 1" >= "1 This is true. Therefore, ( 0 , 1 ) is a solution.
Test the ordered pair (1, 6) Let's test the ordered pair ( 1 , 6 ) .
For the first inequality, we have: 6 < 5 ( 1 ) + 2 6 < 7 This is true. For the second inequality, we have: = "$\frac{1}{2}(1) + 1"> 6 ">= "$\frac{1}{2}(1) + 1 = "0.5 + 1"> 6" >= "0.5 + 1 = "1.5"> 6" >= "1.5 This is true. Therefore, ( 1 , 6 ) is a solution.
Final Answer Therefore, two ordered pairs that make both inequalities true are ( 0 , 1 ) and ( 1 , 6 ) .
Examples
Understanding systems of inequalities is crucial in various real-world scenarios. For instance, consider a business trying to optimize its production. They might have one inequality representing the minimum production required to meet demand and another representing the maximum production capacity. The solution set of this system of inequalities would then represent the feasible production levels that satisfy both demand and capacity constraints. This helps the business make informed decisions about their production planning.
