- A point \((x, y)\) is a solution to the system if it satisfies \(x < 5\), \(y \leq x\), and \(y \geq -3\) simultaneously.
- Geometrically, this represents a region in the Cartesian plane bounded by the lines \(x = 5\), \(y = x\), and \(y = -3\).
- Any point \((x, y)\) that satisfies the compound inequality \(-3 \leq y \leq x < 5\) is a solution.
- To verify a specific point, check if the x-coordinate is less than 5 and the y-coordinate is both greater than or equal to -3 and less than or equal to x.
Explanation
- Understanding the System of Inequalities
To determine if a given point \((x, y)\) is a solution to the system of inequalities, we must check if the point satisfies all three conditions simultaneously:
- \(x < 5\)
- \(y \leq x\)
- \(y \geq -3\)
These inequalities define a region in the Cartesian plane. The boundary lines are \(x = 5\), \(y = x\), and \(y = -3\). Any point \((x, y)\) that lies within this region, satisfying the conditions, is a solution to the system. If even one of these inequalities is false for a point, then the point is not a solution.
- Analyzing the Solution Region
Let us analyze the region defined by these constraints:
- The first inequality, \(x < 5\), means all valid points must lie strictly to the left of the vertical line \(x = 5\).
- The second inequality, \(y \leq x\), means points must be on or below the line \(y = x\).
- The third inequality, \(y \geq -3\), means points must be on or above the horizontal line \(y = -3\).
Combining these, we look for points \((x, y)\) where \(-3 \leq y \leq x < 5\). This implies that for a point to be a solution, its \(y\)-coordinate must be at least \(-3\), and its \(x\)-coordinate must be greater than or equal to its \(y\)-coordinate, but always less than \(5\).
- Testing Points and Conclusion
To illustrate, let us test three example points:
- Point A \((0, 0)\):
- \(0 < 5\) (True)
- \(0 \leq 0\) (True)
- \(0 \geq -3\) (True)
All conditions are met; thus, \((0, 0)\) is a solution.
- Point B \((6, 2)\):
- \(6 < 5\) (False)
Point B is not a solution because it fails the first condition.
- Point C \((2, -4)\):
- \(-4 \geq -3\) (False)
Point C is not a solution because it fails the third condition.
A point is a solution if and only if it satisfies all three conditions simultaneously. Based on these criteria, any point \((x, y)\) satisfying \(-3 \leq y \leq x < 5\) is a valid solution.
Examples
In urban planning, inequalities help establish zoning regulations. Suppose a city mandates that a new building must be built at a distance \(x\) from a central hub such that \(x < 5\) km, while its height \(y\) must satisfy \(y \leq x\) (to prevent blocking views relative to distance) and the structural foundation must be at least \(y \geq -3\) meters (depth). By defining these constraints, planners ensure that all approved building sites satisfy the regional safety and aesthetic requirements. Understanding this system allows planners to verify if any proposed location and height \((x, y)\) is permissible.