Which ordered pairs make both inequalities true? Select two options.
[tex]
\begin{array}{l}
y\ \textless \ 5 x+2 \\
y \geq \frac{1}{2} x+1
\end{array}
[/tex]
(-1,3)
(0,2)
Answer (1)
Substitute the x and y values of the ordered pair (-1, 3) into both inequalities and check if they are true. The first inequality is false, so the ordered pair (-1, 3) does not satisfy both inequalities.
Substitute the x and y values of the ordered pair (0, 2) into both inequalities and check if they are true. The first inequality is false, so the ordered pair (0, 2) does not satisfy both inequalities.
Neither of the given ordered pairs satisfy both inequalities.
Explanation
Understanding the Problem We are given two inequalities: y < 5 x + 2 and y ≥ 2 1 x + 1 . We need to check if the given ordered pairs ( − 1 , 3 ) and ( 0 , 2 ) satisfy both inequalities.
Checking (-1, 3) Let's check the ordered pair ( − 1 , 3 ) . Substitute x = − 1 and y = 3 into the inequalities:
For the first inequality, y < 5 x + 2 , we have: 3 < 5 ( − 1 ) + 2 3 < − 5 + 2 3 < − 3 This is false.
For the second inequality, y ≥ 2 1 x + 1 , we have: 3 ≥ 2 1 ( − 1 ) + 1 3 ≥ − 2 1 + 1 3 ≥ 2 1 This is true.
Since the first inequality is false, the ordered pair ( − 1 , 3 ) does not satisfy both inequalities.
Checking (0, 2) Now let's check the ordered pair ( 0 , 2 ) . Substitute x = 0 and y = 2 into the inequalities:
For the first inequality, y < 5 x + 2 , we have: 2 < 5 ( 0 ) + 2 2 < 0 + 2 2 < 2 This is false.
For the second inequality, y ≥ 2 1 x + 1 , we have: 2 ≥ 2 1 ( 0 ) + 1 2 ≥ 0 + 1 2 ≥ 1 This is true.
Since the first inequality is false, the ordered pair ( 0 , 2 ) does not satisfy both inequalities.
Conclusion Neither of the given ordered pairs satisfy both inequalities.
Examples
Understanding inequalities is crucial in various real-life scenarios, such as budgeting and resource allocation. For instance, if you have a limited budget and need to decide how much to spend on different items while ensuring you stay within your budget and meet certain minimum requirements, you're essentially solving a system of inequalities. This helps in making informed decisions to optimize your resources effectively.
