Mr. Hernandez plotted the point $(1,1)$ on Han's graph of $y \leq \frac{1}{2} x+2$. He instructed Han to add a second inequality to the graph that would include the solution $(1,1)$. Which inequality could Han write?
A. $y>2 x+1$
B. $y<2 x-1$
C. $y \geq 2 x+1$
D. $y \leq 2 x-1$
Answer (2)
The inequality that includes the point (1,1) is y ≤ 2 x − 1 , as it is the only option that holds true when substituting the values of x and y. In contrast, the other options do not satisfy the inequality conditions when (1,1) is tested. Therefore, the chosen option is D.
;
Substitute the point (1,1) into each inequality.
Check if the inequality holds true.
2x + 1"> y > 2 x + 1 becomes 3"> 1 > 3 , which is false.
y < 2 x − 1 becomes 1 < 1 , which is false.
y ≥ 2 x + 1 becomes 1 ≥ 3 , which is false.
y ≤ 2 x − 1 becomes 1 ≤ 1 , which is true.
The correct inequality is y ≤ 2 x − 1 .
Explanation
Understanding the Problem We are given that the point ( 1 , 1 ) lies on the graph of y ≤ f r a c 1 2 x + 2 . We need to find another inequality from the given options such that the point ( 1 , 1 ) also satisfies that inequality.
Testing the Inequalities Let's test each of the given inequalities with the point ( 1 , 1 ) , i.e., x = 1 and y = 1 .
Testing Option 1
2x + 1"> y > 2 x + 1 Substituting x = 1 and y = 1 , we get 2(1) + 1"> 1 > 2 ( 1 ) + 1 , which simplifies to 3"> 1 > 3 . This is false.
Testing Option 2
y < 2 x − 1 Substituting x = 1 and y = 1 , we get 1 < 2 ( 1 ) − 1 , which simplifies to 1 < 1 . This is false.
Testing Option 3
y g e q 2 x + 1 Substituting x = 1 and y = 1 , we get 1 g e q 2 ( 1 ) + 1 , which simplifies to 1 g e q 3 . This is false.
Testing Option 4
y ≤ 2 x − 1 Substituting x = 1 and y = 1 , we get 1 ≤ 2 ( 1 ) − 1 , which simplifies to 1 ≤ 1 . This is true.
Final Answer Therefore, the inequality that includes the solution ( 1 , 1 ) is y ≤ 2 x − 1 .
Examples
Imagine you're designing a simple game where players earn points based on certain conditions. The inequalities we've explored here help define those conditions. For instance, a player might only level up if their score ( y ) is less than or equal to twice their level ( x ) minus one ( y ≤ 2 x − 1 ). By understanding how to test points against inequalities, you can create rules that make the game challenging and fair. This concept extends to many real-world scenarios, such as setting performance targets or defining eligibility criteria.
