The following inequalities form a system.
[tex]
\begin{array}{l}
y \leq \frac{2}{3} x+1 \\
y\ \textgreater \ -\frac{1}{4} x+2
\end{array}
[/tex]
Which ordered pair is included in the solution to this system?
(6,-2)
(6,0.5)
(6,5)
(6,8)
Answer (1)
To determine which ordered pair is a solution to the system of inequalities, we substitute the x and y values of each pair into the inequalities.
For ( 6 , − 2 ) : y − F or (6, 0.5) : y
For ( 6 , 5 ) : y − F or (6, 8) : y Therefore, the ordered pair that satisfies both inequalities is ( 6 , 5 ) .
Explanation
Understanding the Problem We are given a system of inequalities and asked to find which of the given ordered pairs is a solution to the system. To do this, we will substitute the x and y values of each ordered pair into the inequalities and check if both inequalities are true.
Testing (6, -2) Let's test the first ordered pair, ( 6 , − 2 ) .
For the first inequality, $y
For the second inequality, -\frac{1}{4}x + 2"> y > − 4 1 x + 2 , we have -\frac{1}{4}(6) + 2"> − 2 > − 4 1 ( 6 ) + 2 , which simplifies to -1.5 + 2"> − 2 > − 1.5 + 2 , or 0.5"> − 2 > 0.5 . This is false.
Since the second inequality is false, ( 6 , − 2 ) is not a solution to the system.
Testing (6, 0.5) Let's test the second ordered pair, ( 6 , 0.5 ) .
For the first inequality, $y
For the second inequality, -\frac{1}{4}x + 2"> y > − 4 1 x + 2 , we have -\frac{1}{4}(6) + 2"> 0.5 > − 4 1 ( 6 ) + 2 , which simplifies to -1.5 + 2"> 0.5 > − 1.5 + 2 , or 0.5"> 0.5 > 0.5 . This is false.
Since the second inequality is false, ( 6 , 0.5 ) is not a solution to the system.
Testing (6, 5) Let's test the third ordered pair, ( 6 , 5 ) .
For the first inequality, $y
For the second inequality, -\frac{1}{4}x + 2"> y > − 4 1 x + 2 , we have -\frac{1}{4}(6) + 2"> 5 > − 4 1 ( 6 ) + 2 , which simplifies to -1.5 + 2"> 5 > − 1.5 + 2 , or 0.5"> 5 > 0.5 . This is true.
Since both inequalities are true, ( 6 , 5 ) is a solution to the system.
Testing (6, 8) Let's test the fourth ordered pair, ( 6 , 8 ) .
For the first inequality, $y
For the second inequality, -\frac{1}{4}x + 2"> y > − 4 1 x + 2 , we have -\frac{1}{4}(6) + 2"> 8 > − 4 1 ( 6 ) + 2 , which simplifies to -1.5 + 2"> 8 > − 1.5 + 2 , or 0.5"> 8 > 0.5 . This is true.
However, since the first inequality is false, ( 6 , 8 ) is not a solution to the system.
Conclusion Therefore, the only ordered pair that satisfies both inequalities is ( 6 , 5 ) .
Examples
Systems of inequalities are used in various real-world applications, such as in economics to determine the feasible region for production, where constraints on resources like labor and materials are expressed as inequalities. For example, a company might have inequalities representing the maximum amount of steel and labor hours available. The solution to the system of inequalities would then represent the possible production levels that satisfy all constraints. This helps businesses optimize their production plans within the given limitations.
