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 equation could Miguel 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)
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 the point ( 1 , 1 ) and the inequality y ≤ 2 1 x + 2 . We need to find another inequality from the given options that also includes the point ( 1 , 1 ) as a solution.
Solution Plan We will substitute x = 1 and y = 1 into each of the given inequalities and check if the inequality holds true.
Testing Option 1: 2x+1"> y > 2 x + 1 Let's test the first option: 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 Now, let's test the second option: 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 ≥ 2 x + 1 Next, let's test the third option: y ≥ 2 x + 1 . Substituting x = 1 and y = 1 , we get 1 ≥ 2 ( 1 ) + 1 , which simplifies to 1 ≥ 3 . This is false.
Testing Option 4: y ≤ 2 x − 1 Finally, let's test the fourth option: y ≤ 2 x − 1 . Substituting x = 1 and y = 1 , we get 1 ≤ 2 ( 1 ) − 1 , which simplifies to 1 ≤ 1 . This is true.
Conclusion Since the inequality y ≤ 2 x − 1 holds true for the point ( 1 , 1 ) , it is the correct answer.
Examples
When designing a system with multiple constraints, like in engineering or economics, you often need to find solutions that satisfy all conditions simultaneously. For example, in structural engineering, a bridge must support a certain load ( y ) while using a limited amount of material ( x ). If one condition is y ≤ 2 x − 1 , it means the load capacity must be less than or equal to twice the material used minus one unit. Checking if a specific design (represented by a point ( x , y ) ) meets this condition is a direct application of this type of inequality.
To include the point (1,1) on Han's graph of the inequality, the only inequality that holds true is D, which states y ≤ 2 x − 1 . Therefore, Mr. Hernandez can write this inequality. The correct choice is D.
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