Which ordered pair makes both inequalities true?
$\begin{array}{l}
y>-2 x+3 \\
y \leq x-2
\end{array}$
A. $(0,0)$
B. $(0,-1)$
C. $(1,1)$
D. $(3,0)$
Answer (1)
Substitute each ordered pair into both inequalities.
Check if both inequalities are true for the ordered pair.
( 0 , 0 ) : 3"> 0 > 3 (False), 0 ≤ − 2 (False).
( 0 , − 1 ) : 3"> − 1 > 3 (False), − 1 ≤ − 2 (False).
( 1 , 1 ) : 1"> 1 > 1 (False), 1 ≤ − 1 (False).
( 3 , 0 ) : -3"> 0 > − 3 (True), 0 ≤ 1 (True).
The ordered pair that satisfies both inequalities is ( 3 , 0 ) .
Explanation
Problem Analysis We need to find an ordered pair ( x , y ) that satisfies both inequalities:
-2x + 3"> y > − 2 x + 3
y ≤ x − 2
We will test each ordered pair to see if it satisfies both inequalities.
Testing (0,0) Test the ordered pair ( 0 , 0 ) :
For the first inequality, -2(0) + 3 \Rightarrow 0 > 3"> 0 > − 2 ( 0 ) + 3 ⇒ 0 > 3 , which is false.
For the second inequality, 0 ≤ 0 − 2 ⇒ 0 ≤ − 2 , which is false.
Since both inequalities are false, ( 0 , 0 ) is not the solution.
Testing (0,-1) Test the ordered pair ( 0 , − 1 ) :
For the first inequality, -2(0) + 3 \Rightarrow -1 > 3"> − 1 > − 2 ( 0 ) + 3 ⇒ − 1 > 3 , which is false.
For the second inequality, − 1 ≤ 0 − 2 ⇒ − 1 ≤ − 2 , which is false.
Since both inequalities are false, ( 0 , − 1 ) is not the solution.
Testing (1,1) Test the ordered pair ( 1 , 1 ) :
For the first inequality, -2(1) + 3 \Rightarrow 1 > -2 + 3 \Rightarrow 1 > 1"> 1 > − 2 ( 1 ) + 3 ⇒ 1 > − 2 + 3 ⇒ 1 > 1 , which is false.
For the second inequality, 1 ≤ 1 − 2 ⇒ 1 ≤ − 1 , which is false.
Since both inequalities are false, ( 1 , 1 ) is not the solution.
Testing (3,0) Test the ordered pair ( 3 , 0 ) :
For the first inequality, -2(3) + 3 \Rightarrow 0 > -6 + 3 \Rightarrow 0 > -3"> 0 > − 2 ( 3 ) + 3 ⇒ 0 > − 6 + 3 ⇒ 0 > − 3 , which is true.
For the second inequality, 0 ≤ 3 − 2 ⇒ 0 ≤ 1 , which is true.
Since both inequalities are true, ( 3 , 0 ) is the solution.
Conclusion Therefore, the ordered pair that makes both inequalities true is ( 3 , 0 ) .
Examples
Understanding inequalities helps in various real-life scenarios, such as budgeting. For example, if you want to spend less than a certain amount on groceries ( y ) and you know the cost of each item ( x ), you can use inequalities to determine how many of each item you can buy. Similarly, in business, inequalities can help determine the optimal production levels to maximize profit while staying within resource constraints.
