To solve the given inequalities, we need to handle each inequality separately. For the first inequality, \(-7x - 6 > -55\), we will isolate \(x\) by adding 6 to both sides and then dividing by -7, remembering to reverse the inequality sign. For the second inequality, \(-2x + 3 < 5\), we will isolate \(x\) by subtracting 3 from both sides and then dividing by -2, again reversing the inequality sign. Finally, we will combine the solutions using the logical "or" operator.
The first inequality is \(-7x - 6 > -55\). To solve for \(x\), we first add 6 to both sides:
\[
-7x > -49
\]
Next, divide both sides by \(-7\), remembering to reverse the inequality sign:
\[
x < 7
\]
The second inequality is \(-2x + 3 < 5\). Subtract 3 from both sides:
\[
-2x < 2
\]
Now, divide both sides by \(-2\), again reversing the inequality sign:
\[
x > -1
\]
The solutions to the inequalities are \(x < 7\) and \(x > -1\). Since the original problem uses "or," we combine these solutions:
\[
x < 7 \quad \text{or} \quad x > -1
\]
This simplifies to:
\[
-1 < x < 7
\]
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