Solve the following inequalities:
To solve the system of linear inequalities \(6 - 3x \geq 4 - 2x > 5 - 4x\), we split it into two separate inequalities and solve for \(x\).
First, solve the left part: \(6 - 3x \geq 4 - 2x\).
Subtract \(4\) from both sides: \(2 - 3x \geq -2x\).
Add \(3x\) to both sides: \(2 \geq x\), or \(x \leq 2\).
Second, solve the right part: \(4 - 2x > 5 - 4x\).
Add \(4x\) to both sides: \(4 + 2x > 5\).
Subtract \(4\) from both sides: \(2x > 1\), which simplifies to \(x > 0.5\).
Combining these two results, we find that the variable \(x\) must satisfy \(0.5 < x \leq 2\).
To solve the system \(2c + 50 < 7c + 2 + 3c\) and \(5 - 4c < 53\), we solve each independently and find their intersection.
First inequality: \(2c + 50 < 10c + 2\).
Subtract \(2c\) from both sides: \(50 < 8c + 2\).
Subtract \(2\) from both sides: \(48 < 8c\).
Divide by \(8\): \(6 < c\), which means \(c > 6\).
Second inequality: \(5 - 4c < 53\).
Subtract \(5\) from both sides: \(-4c < 48\).
Divide by \(-4\) (recalling to reverse the inequality sign): \(c > -12\).
The solution for the system is the set of values that satisfy both \(c > 6\) and \(c > -12\). Since any number greater than \(6\) is also greater than \(-12\), the solution is \(c > 6\).
Linear inequalities are essential for resource management and decision-making. For instance, imagine a delivery company planning a route. They have a fuel budget constraint (e.g., \(F < 500\) liters) and a time window constraint (e.g., \(T > 5\) hours). By setting up a system of linear inequalities, the planners can identify the range of possible speeds and distances that ensure the delivery remains both cost-effective and timely, ensuring efficient operations.