Absolute value of x raised to the power of 2 +4 less than 5

To solve the inequality:

\[
|x|^2 + 4 < 5
\]

### Step-by-Step Solution:

1. **Simplify the inequality:** Since \(|x|^2 = x^2\), we rewrite the inequality as:
   \[
   x^2 + 4 < 5
   \]

2. **Isolate \(x^2\):** Subtract 4 from both sides:
   \[
   x^2 < 1
   \]

3. **Solve for \(x\):** Take the square root of both sides. Remember that when you take the square root of both sides of an inequality, you must consider both the positive and negative roots:
   \[
   -1 < x < 1
   \]

### Solution:
The solution to the inequality is:
\[
x \in (-1, 1)
\]

This means that \(x\) must be between \(-1\) and \(1\) (not including \(-1\) and \(1\)) for the inequality to hold.

Would you like a more detailed explanation, or do you have any questions?

### Related Questions:
1. How do absolute values affect inequalities differently from normal expressions?
2. What changes if the inequality symbol is \( \leq \) instead of \( < \)?
3. How would you solve \(|x|^3 + 4 < 5\)?
4. What if the inequality was \(x^2 + 4 > 5\)?
5. How do you graph solutions to inequalities like this one?

**Tip:** Always remember to consider both positive and negative solutions when solving inequalities involving squares and square roots.

Absolute value of x raised to the power of 2 + 4 less than 5

Learn how to solve the inequality |x|^2 + 4 < 5. This step-by-step solution explains isolating the quadratic term, simplifying absolute values, and solving for x in the range -1 < x < 1. Ideal for Grades 8-10 students studying inequalities and absolute values.

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