Does $(1,6)$ make the inequality $y < x + 2$ true?
We are given a point \((x, y) = (1, 6)\) and a linear inequality \(y < x + 2\). Our objective is to determine whether the coordinates of the point satisfy this inequality by substituting them into the expression and checking the validity of the resulting statement.
We substitute \(x = 1\) and \(y = 6\) into the given inequality \(y < x + 2\). Replacing the variables with their respective values, we get:
\[6 < 1 + 2\]
Next, we simplify the right-hand side of the inequality:
\[1 + 2 = 3\]
This reduces our inequality to:
\[6 < 3\]
We compare the left-hand side and the right-hand side of the inequality. The value \(6\) is not less than \(3\). Since the statement \(6 < 3\) is mathematically false, the point \((1, 6)\) does not satisfy the inequality \(y < x + 2\). Thus, the correct answer is 'no'.
Determining whether a coordinate pair satisfies an inequality is fundamental in identifying points within specific regions of a coordinate plane. For instance, if you are designing a game where an object must stay within a designated safe zone defined by \(y < x + 2\), you can use this method to check if a specific position \((x, y)\) is within those bounds. This logic is widely applied in geometry, data science, and optimization to classify data points into categories based on boundary constraints.