Which ordered pair makes both inequalities true?
[tex]
\begin{array}{l}
y \leq-x+1 \\
y\ \textgreater \ x
\end{array}
[/tex]
Answer (1)
Test the ordered pair (0, 0) in both inequalities: 0 and 0"> 0 > 0 . The first is true, but the second is false.
Test the ordered pair (1, 2) in both inequalities: 2 and 1"> 2 > 1 . The first is false, but the second is true.
Test the ordered pair (0, 1) in both inequalities: 1 and 0"> 1 > 0 . Both are true.
Test the ordered pair (1, 0) in both inequalities: 0 and 1"> 0 > 1 . The first is true, but the second is false.
The ordered pair that satisfies both inequalities is ( 0 , 1 ) .
Explanation
Understanding the problem We are given two inequalities: y and x"> y > x We need to find an ordered pair ( x , y ) that satisfies both inequalities simultaneously. We will test the given ordered pairs to see which one satisfies both inequalities.
Testing (0, 0) Let's test the ordered pair ( 0 , 0 ) :
For the first inequality, y 0 ≤ − 0 + 1 ⇒ 0 F or t h eseco n d in e q u a l i t y , y > x : 0"> 0 > 0 T hi s i s f a l se . T h ere f ore , (0, 0) d oes n o t s a t i s f y b o t hin e q u a l i t i es .3. T es t in g ( 1 , 2 ) L e t ′ s t es tt h eor d ere d p ai r (1, 2) : F or t h e f i rs t in e q u a l i t y , y 2 \leq -1 + 1 \Rightarrow 2 \leq 0 This is false. Therefore, ( 1 , 2 ) does not satisfy both inequalities.
Testing (0, 1) Let's test the ordered pair ( 0 , 1 ) :
For the first inequality, y 1 ≤ − 0 + 1 ⇒ 1 ≤ 1 T hi s i s t r u e . F or t h eseco n d in e q u a l i t y , y > x : 0"> 1 > 0 T hi s i s t r u e . T h ere f ore , (0, 1) s a t i s f i es b o t hin e q u a l i t i es .5. T es t in g ( 1 , 0 ) L e t ′ s t es tt h eor d ere d p ai r (1, 0) : F or t h e f i rs t in e q u a l i t y , y 0 \leq -1 + 1 \Rightarrow 0 \leq 0 x$:"> This is true.
For the second inequality, $y > x$: 0 > 1 This is false. Therefore, ( 1 , 0 ) does not satisfy both inequalities.
Conclusion The ordered pair ( 0 , 1 ) satisfies both inequalities.
Examples
Imagine you're designing a simple game where a character can only move within certain boundaries. The inequalities y and x"> y > x could represent those boundaries. For example, y might mean the character can't go too high relative to its horizontal position, and x"> y > x might mean it has to stay above a certain diagonal line. Finding the ordered pair (0, 1) that satisfies both inequalities tells you one possible position where the character is allowed to be, ensuring it stays within the game's rules.
