Question

Determine the solution set of $2x\gt -8$ and $-5x+7=12$ (1 point)
$\{ -4\} $
$\{ x\vert x\gt -4\} $
$\{ x\vert x\gt -1\} $
$\{ -1\} $

Answer 1

To solve the inequalities and equations provided, we will address each part separately.

1. **Solve the inequality \(2x > -8\):**

Start by dividing both sides by 2:
\[
x > -4
\]
This means \(x\) can be any number greater than -4.

2. **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:
\[
\{ x \mid x > -4 \}
\]