To solve the inequalities and equations provided, we will address each part separately.
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Solve the inequality \(2x > -8\):
Start by dividing both sides by 2: \[ x > -4 \] This means \(x\) can be any number greater than -4.
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Solve the equation \(-5x + 7 = 12\):
First, isolate the term with \(x\): \[ -5x = 12 - 7 \] Simplifying the right side: \[ -5x = 5 \] Now, divide by -5 to solve for \(x\): \[ x = -1 \]
Now we have the solution set for both the inequality and the equation:
- The solution set to the inequality \(2x > -8\) is \( { x \mid x > -4 } \).
- The solution to the equation \(-5x + 7 = 12\) is \({ -1 }\).
Since we need to determine the solution set that works for both, we check \(x = -1\):
- Since \(-1\) is greater than \(-4\), it satisfies the inequality \(2x > -8\), meaning it is a valid solution.
Looking at all options given:
- \({ -4} \) does not make sense, as -4 is not greater than -4.
- \({ x \mid x > -4 }\) is valid, as we can have all numbers greater than -4.
- \({ x \mid x > -1 }\) is not valid, as -1 is part of our equation's solution.
- \({ -1 } \) is a specific solution.
In summary, because \(-1\) is indeed greater than -4 and satisfies both conditions, the implication is that:
- We recognize \(x = -1\) is a solution to the equation, while \(x\) being described as greater than -4 means we have a broader set.
Thus, the best representation of the solution set that encompasses both conditions stated is \(x > -4\).
Therefore, the answer that most accurately encompasses our findings is:
\[ { x \mid x > -4 } \]
The correct answer from the choices is: \[ { x \mid x > -4 } \]