Solve the equation. Be sure to check for extraneo $$ |2 k+1| \geq 9 $$ Interval Notation: typeyour answer...

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UpStudy AI · Accepted Answer

To solve the inequality $$ |2k + 1| \geq 9 $$, we'll follow these steps:

1. **Understand the Absolute Value Inequality:**
   
   The inequality $$ |2k + 1| \geq 9 $$ means that the expression inside the absolute value, $$ 2k + 1 $$, is either:
   
   - Greater than or equal to 9, or
   - Less than or equal to -9.

2. **Set Up the Two Scenarios:**
   
   a. **First Scenario:**
      $$
      2k + 1 \geq 9
      $$
      
      Subtract 1 from both sides:
      $$
      2k \geq 8
      $$
      
      Divide both sides by 2:
      $$
      k \geq 4
      $$
   
   b. **Second Scenario:**
      $$
      2k + 1 \leq -9
      $$
      
      Subtract 1 from both sides:
      $$
      2k \leq -10
      $$
      
      Divide both sides by 2:
      $$
      k \leq -5
      $$

3. **Combine the Solutions:**
   
   The solutions from both scenarios are:
   $$
   k \geq 4 \quad \text{or} \quad k \leq -5
   $$
   
   This means that $$ k $$ is in one of two intervals:
   
   - $$ k $$ is greater than or equal to 4.
   - $$ k $$ is less than or equal to -5.

4. **Express in Interval Notation:**
   
   Combining both intervals, the solution in interval notation is:
   $$
   (-\infty, -5] \cup [4, \infty)
   $$

**Final Answer:**
$$
(-\infty, -5] \cup [4, \infty)
$$
The solution in interval notation is $$ (-\infty, -5] \cup [4, \infty) $$.

Solve the equation. Be sure to check for extraneo \( |2 k+1| \geq 9 \) Interval Notation: typeyour answer...

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