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 each ordered pair into both inequalities.
Check if both inequalities are true for the given ordered pair.
If both inequalities are true, then the ordered pair is a solution.
Neither of the given ordered pairs satisfy both inequalities.
Explanation
Understanding the Problem We are given two inequalities: y < 5 x + 2 and = ""> y " >= " \frac{1}{2}x + 1 . W e n ee d t oc h ec k w hi c h o f t h e g i v e n or d ere d p ai rs , (-1, 3) an d (0, 2)$, satisfy both inequalities.
Checking (-1, 3) in the first inequality Let's check the ordered pair ( − 1 , 3 ) . Substitute x = − 1 and y = 3 into the first inequality: 3 < 5 ( − 1 ) + 2 3 < − 5 + 2 3 < − 3 This is false.
Conclusion for (-1, 3) Since the first inequality is false for ( − 1 , 3 ) , we don't need to check the second inequality. The ordered pair ( − 1 , 3 ) does not satisfy both inequalities.
Checking (0, 2) in the first inequality Now let's check the ordered pair ( 0 , 2 ) . Substitute x = 0 and y = 2 into the first inequality: 2 < 5 ( 0 ) + 2 2 < 0 + 2 2 < 2 This is false.
Conclusion for (0, 2) Since the first inequality is false for ( 0 , 2 ) , we don't need to check the second inequality. The ordered pair ( 0 , 2 ) does not satisfy both inequalities.
Final Answer Neither of the given ordered pairs satisfy both inequalities.
Examples
Understanding inequalities helps in various real-life situations, such as budgeting (spending less than a certain amount) or meeting minimum requirements (like scoring above a certain threshold in a test). In this case, we checked if certain points satisfied given conditions, similar to checking if a particular plan fits within certain constraints.
