Question
Real Tutor Solution
Tutor-Verified Answer
Answer
The solution is all real numbers less than -4 or greater than or equal to 1. In symbols, \( x < -4 \) or \( x \geq 1 \).
Solution
To solve the inequalities \( -2x - 7 > 1 \) **or** \( x - 2 \geq -1 \), we'll solve each inequality separately and then combine their solutions.
### 1. Solving \( -2x - 7 > 1 \)
**Step 1:** Add 7 to both sides to isolate the term with \( x \):
\[
-2x - 7 + 7 > 1 + 7 \\
-2x > 8
\]
**Step 2:** Divide both sides by -2. **Remember:** Dividing or multiplying both sides of an inequality by a negative number reverses the inequality sign.
\[
x < \frac{8}{-2} \\
x < -4
\]
**Solution for the first inequality:**
\[
x < -4
\]
### 2. Solving \( x - 2 \geq -1 \)
**Step 1:** Add 2 to both sides to solve for \( x \):
\[
x - 2 + 2 \geq -1 + 2 \\
x \geq 1
\]
**Solution for the second inequality:**
\[
x \geq 1
\]
### Combining the Solutions
Since the original problem uses **"or"**, the final solution includes all \( x \) that satisfy **either** of the two inequalities.
\[
x < -4 \quad \text{**or**} \quad x \geq 1
\]
**In interval notation:**
\[
(-\infty, -4) \cup [1, \infty)
\]
### Final Answer
All real numbers below –4 or at least 1. In symbols,
x < −4 ∨ x ≥ 1
Or in interval notation:
(−∞, −4) ∪ [1, ∞)
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To solve the inequalities, start with the first one: \( -2x - 7 > 1 \). Add 7 to both sides to get \( -2x > 8 \). Then, divide by -2, remembering to flip the inequality sign, resulting in \( x < -4 \). Now for the second inequality: \( x - 2 \geq -1 \). Add 2 to both sides to get \( x \geq 1 \). Putting it all together, the solution is \( x < -4 \) or \( x \geq 1 \). In interval notation, that’s \( (-\infty, -4) \cup [1, \infty) \). Now you have the complete solution set!
