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The solution to \( |x| < 4 \) is \( -4 < x < 4 \).
Solution
To solve the inequality \( |x| < 4 \), we need to understand what the absolute value means. The expression \( |x| < 4 \) indicates that the distance of \( x \) from 0 is less than 4.
This can be expressed as a compound inequality:
\[
-4 < x < 4
\]
### Step-by-step explanation:
1. **Understanding Absolute Value**: The absolute value \( |x| \) represents the distance of \( x \) from 0 on the number line. Therefore, \( |x| < 4 \) means that \( x \) is within 4 units of 0.
2. **Breaking Down the Inequality**: The inequality \( |x| < 4 \) can be rewritten as two separate inequalities:
- \( x < 4 \)
- \( x > -4 \)
3. **Combining the Inequalities**: We can combine these two inequalities into one compound inequality:
\[
-4 < x < 4
\]
### Conclusion:
The solution to the inequality \( |x| < 4 \) is the interval:
\[
(-4, 4)
\]
This means that \( x \) can take any value between -4 and 4, not including -4 and 4 themselves.
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Simplify this solution Extra Insights
Did you know that the inequality \( |x| < 4 \) translates to saying that \( x \) is confined to a very cozy "house" on the number line? Specifically, it means \( x \) can only take on values between -4 and 4 (exclusive). So, you can think of it as saying, "Stay within the bounds, but not touching the walls!" When solving this inequality, a common mistake is treating the absolute value as a simple number. Remember to split it into two inequalities: \( -4 < x < 4 \). Always double-check your range! If you cross the walls of the absolute value, you’ll find you’ve strayed out of bounds!
