X over 4 greater than or equal to negative 8
The Correct Answer and Explanation is:
The inequality given is:
[
\frac{x}{4} \geq -8
]
Step-by-Step Solution:
- Multiply both sides by 4:
To eliminate the denominator of 4 on the left-hand side, we multiply both sides of the inequality by 4. It’s important to note that multiplying or dividing by a positive number does not change the direction of the inequality.
[
\frac{x}{4} \times 4 \geq -8 \times 4
]
This simplifies to:
[
x \geq -32
]
Final Answer:
The solution to the inequality is:
[
x \geq -32
]
This means that the value of (x) must be greater than or equal to (-32).
Explanation:
- Understanding the inequality: The inequality (\frac{x}{4} \geq -8) suggests that the value of (x) divided by 4 is greater than or equal to (-8). The goal is to isolate (x) and find the range of values it can take to satisfy the inequality.
- Multiplying by 4: By multiplying both sides of the inequality by 4, we aim to eliminate the denominator and simplify the inequality. This step is crucial because it removes the fraction, making it easier to understand and solve. Since we are multiplying by a positive number (4), the direction of the inequality stays the same.
- Interpreting the result: After simplifying, we find that (x \geq -32). This tells us that any value of (x) that is greater than or equal to (-32) will satisfy the inequality. In interval notation, the solution is written as ([-32, \infty)), meaning (x) can be any number starting from (-32) and extending to positive infinity.
- Real-world context: If this inequality represented a real-world situation, such as a minimum acceptable score or amount, it means that the value of (x) (perhaps the score or quantity) cannot be less than (-32). Therefore, any value of (x) from (-32) onward would be valid.