Question
6 Fill in the Blank 1 point Solve the equation. Be sure to check for extraneous soluti \( |4 m-1|-5 \leq 0 \) Interval Notation: type your answer...
Ask by Young Mcdonald. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solution in interval notation is \([-1, \frac{3}{2}]\).
Solution
To solve the inequality \( |4m - 1| - 5 \leq 0 \):
1. **Isolate the absolute value:**
\[
|4m - 1| \leq 5
\]
2. **Rewrite the inequality without the absolute value:**
\[
-5 \leq 4m - 1 \leq 5
\]
3. **Solve the compound inequality:**
- Add 1 to all parts:
\[
-4 \leq 4m \leq 6
\]
- Divide by 4:
\[
-1 \leq m \leq \frac{3}{2}
\]
4. **Check for extraneous solutions:**
Since the operations performed are valid and don't introduce any extraneous solutions, all solutions are valid.
**Interval Notation:**
\[
[-1, \frac{3}{2}]
\]
**Answer:** \([-1, \frac{3}{2}]\)
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Simplify this solution The Deep Dive
To solve the inequality \( |4m - 1| - 5 \leq 0 \), first isolate the absolute value: \[ |4m - 1| \leq 5. \] This means the expression inside the absolute value can range between -5 and 5: \[ -5 \leq 4m - 1 \leq 5. \] By solving the left side: \[ -5 + 1 \leq 4m \implies -4 \leq 4m \implies -1 \leq m. \] For the right side: \[ 4m - 1 \leq 5 \implies 4m \leq 6 \implies m \leq \frac{6}{4} \implies m \leq 1.5. \] Combining these inequalities gives: \[ -1 \leq m \leq 1.5. \] In interval notation, this is expressed as: \[ [-1, 1.5]. \] Now to check for extraneous solutions, both endpoints are valid solutions to the original inequality; therefore, there are no extraneous solutions.
