$$ k-3

Question

$$ k-3<-7 $$ or $$ k+5 \geq 8 $$

UpStudy AI · Accepted Answer

Let's solve the compound inequality:

$$
k - 3 < -7 \quad \text{or} \quad k + 5 \geq 8
$$

**1. Solve $$ k - 3 < -7 $$:**

$$
k - 3 < -7 \
\text{Add } 3 \text{ to both sides:} \
k < -7 + 3 \
k < -4
$$

**2. Solve $$ k + 5 \geq 8 $$:**

$$
k + 5 \geq 8 \
\text{Subtract } 5 \text{ from both sides:} \
k \geq 8 - 5 \
k \geq 3
$$

**Combined Solution:**

The solution to the compound inequality is all real numbers $$ k $$ that satisfy either $$ k < -4 $$ **or** $$ k \geq 3 $$.

**In Interval Notation:**

$$
(-\infty, -4) \cup [3, \infty)
$$

**Graphical Representation:**

```
<---|---------|=========>
    -4        3
```

- The open circle at -4 indicates that $$ k $$ can be any number less than -4.
- The closed circle at 3 indicates that $$ k $$ can be 3 or any number greater than 3.

**Final Answer:**

All real numbers k with k < –4 or k ≥ 3. In interval notation:
 (–∞, –4) ∪ [3, ∞)
The solution to the inequality $$ k - 3 < -7 $$ or $$ k + 5 \geq 8 $$ is all real numbers $$ k $$ such that $$ k < -4 $$ or $$ k \geq 3 $$. In interval notation, this is written as $$ (-\infty, -4) \cup [3, \infty) $$.

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