- Solve the first inequality \(5x + 7 \leq -3\) to get \(x \leq -2\).
- Solve the second inequality \(3x - 4 \geq 11\) to get \(x \geq 5\).
- Combine these solutions using the "or" operator, resulting in the union \(x \leq -2 \quad \text{or} \quad x \geq 5\).
- The correct choice is: \(\boxed{x \leq -2 \quad \text{or} \quad x \geq 5}\).
Explanation
- Analyzing the Problem
The given compound inequality is \(5x + 7 \leq -3 \quad \text{or} \quad 3x - 4 \geq 11\). The word "or" signifies that the solution to this compound inequality is the union of the solution sets of the two individual inequalities. We need to solve each linear inequality separately to find the values of \(x\) that satisfy the condition.
- Solving the First Inequality
First, we solve the inequality \(5x + 7 \leq -3\):
Subtract 7 from both sides:
\(5x \leq -3 - 7\)
\(5x \leq -10\)
Divide both sides by 5:
\(x \leq \frac{-10}{5}\)
\(x \leq -2\)
- Solving the Second Inequality
Next, we solve the inequality \(3x - 4 \geq 11\):
Add 4 to both sides:
\(3x \geq 11 + 4\)
\(3x \geq 15\)
Divide both sides by 3:
\(x \geq \frac{15}{3}\)
\(x \geq 5\)
- Conclusion and Selecting the Correct Option
Combining the two results, we get the solution set: \(x \leq -2\) or \(x \geq 5\). Comparing this result with the given choices, the correct option is \(x \leq -2 \quad \text{or} \quad x \geq 5\).
Examples
Understanding compound inequalities is useful in scenarios where you have a "safety range" or a "boundary condition." For example, if a machine in a factory needs to operate outside a certain temperature range to prevent damage, or if you are measuring the acceptable tolerance of a component's dimensions, you are essentially solving compound inequalities to determine the valid working zones for your project. Knowing how to combine these ranges helps in defining clear operating limits for engineering, logistics, and resource management.